If the design of the house permits it, i.e., has an attic with access to it, a whole house attic fan is worth considering. If not or if this is a rental that would preclude this, then upstairs window fan would be the way to go. Window fans will provide more cooling for your energy buck than table top fans though those can be useful for localized personal cooling.

A large attic fan with all windows slightly open is a fabulous cooler. It is loud, though. A quieter, constantly running solar powered window fan and/or solar powered attic-vent fan could reduce electric grid usage without sacrificing too much of your quietness.

If you do not have tree shade, wide porches or window awnings, then you could shade the areas near your sunniest windows with one of those triangular sail-cloth-looking thingies.

If your humidity is fairly low, you can place bowls of ice on a tabletop and/or hang wet sheets at the windows (or dry your clothes indoors) to reduce indoor temps. Doesn’t work in humid days/climes.

In the Old South Delta country, people built cabins with two rooms on either side of a long roofed-over but otherwise open passage that permitted the prevailing winds to pass through, creating a tunnel effect breeze. It was called a ‘dog trot’ style of house. You could create a similar effect with a toolshed or garden shed fairly close to the house and wide overhanging eaves on both house and shed

In a humid summer, having a dehydrator inside is almost as good as having A/C, which acts as a dehydrator /condensor. Keeping one room drier in summer is like keeping one room warmer in winter.
It gives a place to hang out when the muggy buggles are getting to you. Or install an A/C window unit in just one room and run it intermittently to provide a heat relief refuge.

Digging out a root cellar beneath the house can be another option. A hangout in summer, a food storage and canned goods pantry all winter, and a defense against cold snaps in spring for germinating plant starts.

Leaving the windows open at night so cooler air comes in, then closing windows during the morning hours to keep the cool air in, and reopening them in the afternoon to let hot air flow out (cf. Teresa’s post on ‘the window dance’) helps regulate indoor temps. Heat reflective thermal shades are good for keeping out winter cold and summer heat.

Old-time farmhouse dairies used to be situated on the North side of the barn or house, and oriented with airflow from low in the Southeast to higher in the Northwest, not letting much (if any) sunshine in from East and West. Then they installed an unglazed tile floor and same halfway up the walls. You damp these down in the cool of the morning and force air flow across the wet room. As it dries out, it cools down to keep the milk fresh. You might could find old timey floor plans and building designs for dairies in historical societies that could help you do something like.

Cooking outside on a barbecue brick pit or grill instead of the range and stovetop can help prevent heating up the interior.

I spend a LOT of time thinking about cooling a house naturally.
First: do you own?
Second: are you staying put?

This determines what you can do.
I've written extensively on the subject (https://peschelpress.com/fed-safe-and-sheltered/) but here are the basics.

If you own, figure out your sun arc over the house. It will take a full year to learn the details of the sun's path overhead. In general though, plant evergreens along the north side of the property to shield you from winter winds. Next, plant deciduous trees on the other three sides. Size of tree is determined by how much space you've got. Don't plant trees that grow to 100 feet at maturity near powerlines or in tight spaces where the tree can't spread. This is a long-term project. You won't see results for years but they WILL show up.

Install white window shades or blinds (room darkening is best) to block direct solar gain when it's hot. Close the shade or blinds and you reflect sunlight back outside. Open and close as needed to air out the house. Shades and blinds do double-duty in trapping heat in the winter.

Fans do work (moving air) but they're best for catching cooler, night air. Leave windows open at night with fans blowing IN on the ground floor and OUT on the upper floor, forcing natural air movement. Close up in the morning, close your drapes and trap your coolness. Look for other drafting possibilities: fireplace vents, attic accesses, basement windows and such. You have to open these vents when it's cool and close them when it gets hot otherwise your house will get hotter than ever.

Ceiling fans can work even with very low ceilings. We have 7 and 1/2 foot ceilings and have a ceiling fan in each room. Look for low-profile fans. You may be able to find ones that will work. You may have to special order them. At a minimum, you can install this type of fan in two places where it really matters AND you won't walk into the fan: over the dining room table and over a bed. Fans should always be as close to the center of the room as possible but if it's a choice between no fan and a fan over the bed or the table, take the fan. Get the biggest blades that will fit in the room. Bigger blades move more air. A ceiling fan makes it feel cooler than it is and you aren't cooling empty rooms.

The goal is always to keep the heat out! More insulation in your attic will go a long way to keeping you more comfortable in winter and summer. Add ridge vents to your roof when you reroof and get the lightest roof color possible. A cooler attic makes for a cooler house. It's much easier to trap heat than to repel it.

Awnings can be great. Vines growing up the screens on the side of porches. More grass or planted areas close to the house and less asphalt. White houses repel heat; black painted houses suck it up so choose a light house color.

The drawback is it takes time and effort and paying attention. Someone has to do the window dance twice a day. But it is doable. It will make you more comfortable, save you money, and keep you in better tune with your house.

If your attic is accessible, you can install attic insulation yourself IF you can reach and you're meticulous. Any gap in insulation damages the magic.

Don't insulate in the summer. It's too darn hot and you'll fry.
Pick a cool, cloudy day and put on your coverall or boiler suit. Wear gloves. Clean out the bays of crap and install the insulation batts like the directions tell you. Don't block any vents. Then install layer number two crosswise. You can add a third layer if you like.
One oddity of our house was the second floor addition which did NOT cover the entire first floor. That is, it's a room atop part of the first floor. There was a few inches of rock wool in our bays under randomly thrown-about batts (the previous owners had zero understanding of insulation and heated the attic via heating the house and warm air rising). Bill removed the dead, unusable batts. He then, with a hoe, shoved the rock wool under the second-floor addition, assuming that no insulation had been installed between the two floors when it was added. Some bays had more space than others but all the rock wool went under the addition. It seemed to help.

The main issue with insulating your attic yourself is time and being meticulous. It's a tedious, hot, sweaty, unpleasant job. However, if you can reach an area, you can insulate it. Bill's done two houses now, here and in South Carolina.

You do NOT want to block any vents to avoid moisture damage.

I would like to add that you consider creating a summer kitchen of some sort. I just got mine set up under a roof of shade cloth that is anchored to the north side of my house stretched over an old garage tent frame then staked to the other side with guy ropes. I have a Camp Chef stove set up in the shade and we cook on it all summer as well as do a lot of canning on this stove. It is propane fueled, so you have to keep yourself supplied, but that isn't too hard. One of the beauties of propane is that it burns hotter then natural gas, so things heat up quicker and you can cook faster and use less propane. We got this setup just in time as we had our first 100 degree day of the summer. Looks to be another long and hot one.

It got above 95F this past week, and I finally got out my own fans. Up until now, the natural sump of cooler air in the house has kept things if not comfortable, at least tolerable in the late afternoon. I've also been out at the new place working until 5pm everyday, so I could come home, take a cold shower and crash for a nap.

I put a fan in the kitchen door last night at about 10pm for a few hours to pull in cool evening air, and vent some of the heat out the front door. This morning while I had coffee, I turned it back on. I'll close up the doors when I leave. I expect that should pull the temperature down 5-10 degrees when I get home today.

Even a lazy application like mine can have good affects on your situation.

The rule I've heard on moving air is that you pull air, you don't push it. The illustration was a chain - you can move a chain easily by pulling it, much harder to move by pushing it. It balls up, has little side-jaunts and generally doesn't work well.

One slight more expensive option for solar shielding is to pick up sleeping bags at garage sales or Wal-Mart. They are typically the right size to cover big windows if opened all the way. Thick and insulated, usually with a lighter color on the inside for Summer and flipped a darker side you can use for Winter. Their weight usually keeps them from blowing around in drafts too. I'm got them over all of my windows with just a slim open for the cats to climb up in the front window.