Theme keeps breaking

I recently took over admin duties for an organization called the American Club of Lille. They were using WordPress 2.3 hosted on a server run by one of their members (to which I have FTP access).

As one of the first steps in repairing and updating their site, I updated the WordPress to 3.5. There wasn’t an automatic upgrade button on such an old version, so I did it manually:
-I asked the server owner to back up the databases.
-Made a local copy of all the files on the server.
-Downloaded 3.5 as a zip.
-Deleted wp-admin and wp-includes, uploaded ones from zip.
-Merged wp-content from zip with existing wp-content, overwriting where necessary.
-Uploaded loose files from zip to the directory containing the directories wp-content, wp-admin, etc

Thankfully everything seemed to work right off the bat. Until it didn’t.

No matter which theme I select, it will stay that way from a few minutes up to hours, then will switch to a blank screen and then to the default theme.

I deleted the default theme and now it resets to the TwentyTwelve theme instead. I tried disabling all plugins and nothing changed. I’ve tried automatically reinstalling 3.5 via the button on the dashboard, but no change. I’ve checked the file permissions based on advice from other threads; they all seem to be 755.

If I log into the dashboard while the website is displaying a white page, I see: “ERROR: The themes directory is either empty or doesn’t exist. Please check your installation.”. Once I click to the themes page, everything is fine, and going back to the dashboard shows the TwentyTwelve theme selected.

I haven’t been able to figure out what’s causing the random reset, let alone how to fix it.

Please help me figure out how to get the theme to stay put. If you need any info I didn’t provide, let me know and I’ll be happy to give it.