OK, so here's what I did.
1. I copied my old WinAGI install elsewhere.
2. I uninstalled it. (Guess I can skip these steps on a system that WinAGI isn't already installed on?)
3. I put the new exe and dll in a folder. Not in program files, just wherever the hell I wanted. I also copied the other files from the old install to the new folder.
4. I ran regsvr32 in an admin command prompt to register the dll.
5. I ran WinAGI.
And well, the saving bug is gone! But wait, there's more! I couldn't edit logics anymore, and when compiling the game I got errors about RichEditAGI.ocx. But don't worry, I just registered it too with regsvr32 (again as admin). Guess you just forgot to include this step, so maybe this will help someone.
Aaaand, I could edit logics normally, and the compiling bug I had encountered is gone. I guess it was related to the saving bug!
Making good progress here I think. I'll experiment and report back. Ironically, these fixes are kind of useless for me, as I don't really intend to do the developing on Windows 7 anyway, but who says fixing bugs is bad? You should have the least buggy software ever (everyone should) and maybe something else got fixed too. So yay.