Is there any secret antidote for clay, or some common (and cheap!) additive I could use?
This post is going to be really long, but this is what I would do.
July in Ohio is really too late to start a garden, it should have been started in the middle of May. I'd call this year a loss, and start digging. My goal would be to turn the garden into a huge compost pile/worm bin. Worms aerate the soil and poop good dirt.
All worms want is food and bedding. If you build it, they will come!!
I would get my hands on any and all organic matter I could and dig it in. (All things in moderation.) I don't know what's common to your area, but go collecting those cheap additives!! Grass clippings, all summer long. (Get your neighbor's grass clippings if possible.) Dig in your kitchen scraps and shredded papers.
Do you live near any place that has livestock and might let you haul away the manure and used bedding? For the next three months, get whatever you can.
I say three months because horse, cow, and chicken manures are considered "hot" and need to age a year. I think if you throw enough carbon in there and keep it wet enough, you could shave that down by a couple months. (This is my personal opinion and may be disputed by others, so YMMV.) After three months, dig in any other manure/bedding you can find. You might double-check my memory, but I think
all other manures are "green" and can be added at any time. Find someone who keeps goats or herds sheep or raises rabbits. Find an alpaca farm. Get the kids a guinea pig!
Ok, that last one might be reaching, but you get the gist. See if the area where you live has a 4H or FFA club. I'm sure those kids would
gladly let you muck out the pens once a week!
More common and cheap additives - If you live close to a coffee shop, convenience store, or restaurant, you could ask them to save their used coffee grounds for you. Take them a couple 5 gallon buckets. (There's a guy on YouTube that hits up his local Starbuck's and collect 40 pounds a week!
.) If you do this, tell them they can save the filter, too. They'll be more likely to help out if it's hassle-free. The filter is biodegradable; it's brown material. One reason this would be
especially good for your garden is that worms, like birds, have gizzards and need to eat "grit" as an aid in digestion. Clay doesn't really have a lot of grit.
Tree leaves this fall. (Your neighbor's tree leaves this fall.
.) If you can, chop them with a mower before you dig them in, they'll break down quicker. (Don't forget the Halloween pumpkins, worms love pumpkin.)
One time at my sister's cabin on the lake, I watched her pull a huge amount of duckweed out of her little dock area. (It was so thick it was binding the boat motor!) It was so green and healthy, grown in the lake water, I wished I could have it for my compost pile!! Quick google search tells me the stuff is invasive almost everywhere. Got a lake handy? (Seriously, she had three big black trash bags full in less than an hour and a half!!
.)
Do you live close to a city that has a free paper shredding day? They have to do
something with all that paper. Make a phone call and see if you can get a couple bags!
Can you find pine needles? I've read that they are good for helping loosen clay soil. They decompose slowly, and the springy quality of them helps the fight the weight of the clay. Some gardeners will say that they'll make your soil acidic, but I think that's been disproved. If you are one of those gardeners, please watch this video.
Definitely, call tree trimming services (or your county road crew) and find out if you can get shredded wood chips. A good time for this is after a storm takes down some trees. If you can find a sawmill or lumber mill, you might be able to get sawdust.
A bale of straw is not too expensive, get a bale and dig it in. Two, if your garden is bigger.
Got any flat pop? ... "Soda," if that's the word you're used to.
How about stale beer, or yesterday's cold coffee? Pour that in there. If you have a freshwater aquarium, pour the water from your water changes on it. Heck, pee in a bucket, and put that in there! (Yes, I'm serious.)
Do this from now until winter, then just let it settle. Since what you are trying to do is make new dirt, I wouldn't worry about pH right now at all. Get all the organic material you can in there, and worry about adjusting your pH early next spring. Think of it like a balanced diet, and everything organic brings something different to the table. Everything will break down at different rates and add tilth to the soil.
https://en.wikipedia.org/wiki/Tilth#Fine-textured.2C_clay_soils
Probably the least "nutritious" thing is the paper, but worms like to use it for bedding.
Since you are least likely to be adding soil amendments over the winter, when it is warm enough for worms to be active, they will eat their bedding. (Above 40 degrees.)
Hope that helps!!
Welcome to the forum!! 
