Okay first off, sorry for the delay in replies. The past week has been... nuts. Second, thank you all for the replies with various tips and tricks to slow/stop/remove the rust inside my frame. Based on your advice, I think what I'll do is get a fridge coil brush to clean up the seat tube as best as I can, then use some of my brass wire brushes and a drill to try and clean up the bottom bracket and headset as best as I can. I'll then use my air compressor to blow out whatever dust and debris I can. I know a lot of people recommended plugging the frame and pouring POR15 into the frame, and I definitely want to try that at some point, but I think I'll try using used ATF instead. I already have some in stock, since I was mixing it with acetone to make a rust-busting solution for use on another, even rustier bike. Once I'm in a better financial position, I'll definitely look into POR15, but right now, I'll just make do with what I have.
For those who were wondering how the bearings rusted up, my guess is that there's so much rust chunks and dust inside the frame, that just moving the bike around as much as I did allowed that junk to work its way inside the bearing cups from the inside. It looks like there's enough space between all the parts for it to do that, at least from what I can tell. Plus, the rust just wiped off the bearing cups and bearings, so it was really getting stuck to the grease, what little was on everything at the time.
One last question: how do you suggest I plug up the frame? I could probably make do with duct/gorilla tape, if I could use a hair dryer or heat gun to peel it off cleanly, but I don't know if there are any other alternatives. Any suggestions for that?