I need to save/compile a few times and a bunch of weird files (like foo.tex.Ak6664) appear in my Dropbox folder each time

I have the same problem. According to the bug report, it seems to be related to a change in the way TeXstudio saves a file. What can you do? Two options:

a) Give up the cloud

... and simply work outside your Dropbox, as the OP suggested. You can still manually copy files into the Dropbox to back them up or share them.

b) Downgrade TeXstudio and keep the cloud

My (hopefully just temporary) solution: Downgrade to TeXstudio 2.10.8, which saves files the old way. I’m using the portable version in order to be able to run it alongside the current one.

Transferring your settings to the portable version is really easy: Find the settings file texstudio.ini and copy it to the directory of the portable version. (TeXstudio needs to be closed while you do that.) I’m not sure if you also need to import the .txsprofile file, if somebody can clarify that, let me know!

Dropbox was creating these additional files (not sure exactly why) due to space capacity constraints. Small files were not causing any issues. Everything works fine on my desktop. Although I now have enough capacity on dropbox, I think it takes dropbox a few days to update.

Edit1:

• It turns out that this is a texstudio bug and not a problem with dropbox (and that my solution is incorrect) - see @UlrikeFischer link in the comments below.

• I plan to just work on my desktop anyway and copy files to dropbox at the end of the day until the bug is fixed (hopefully soon). Alternatively, I'm sure @doncherry solution also works.

Edit2: I stopped working on the desktop and downgraded texstudio like @doncherry suggested.

Version 2.12.0 has solved the issue. The changelog file includes the line

TeXstudio 2.12.0
[...]
- revert to pre 2.11.0 save method on Windows because the new QSaveFile has problems with Dropbox folders

I have checked and it does work for me on Windows 10.