@APowers The way the hinge is shaped, it blocks the screw on the right. There's not much space at all to get anything in. It's hard to see in the picture because the freezer door is blocking it. But I can try a side on approach with an open end wrench. I didn't think to borrow one from work when I tried the other night. (I have basically no tools beyond a hammer and a normal screwdriver. I borrowed the ratchet set from work too.)
@Papa bear You're right. Based on the manual, there should be leveling legs. I also have some leftover shims from when I needed to level a bookcase recently if necessary.
I'll give it another shot this weekend. Maybe if I level it first, try to adjust the hinges again with more tools, and rearrange the fridge with the weight of the door in mind, I can get it fixed. If not, I'll try to seal the gap somehow. Fingers crossed it works.
If none of that helps our energy bill, I might consider a new fridge as
@middo suggested for the long-term energy savings. But that would be a last resort, and I'd probably at least run it by my super/landlord to see if they'd do anything.
At any rate, thanks so much for all your suggestions! I was pretty stuck the other night and now I have some more things to try.