Will changing themes fix the stuff that the last update broke?
No idea at all. ;)
If it's your current theme that's causing your issues then probably. You can always temporarily switch to the Twenty Eleven theme and test. You won't lose anything by trying and you can always switch back.
Start with the easy stuff:
- Deactivate all of your plugins. Really, everyone of them.
- Switch to the Twenty Eleven theme. Someday, probably soon, I'll tell people Twenty Twelve but for now use Twenty Eleven.
- Re-login to your WordPress installation.
- Create a new post and see if the Visual Editor works or not.
If it does work then (and this may be lengthy) leave the theme at Twenty Eleven and re-activate 1 plugin. Clear the cache and cookies and test again.
Rinse, wash, repeat until the problem comes back. Switch the theme back last.
That will tell you if it's a plugin or your theme that's causing you visual editor grief.
If that doesn't work (don't leave out any steps) then give the 3.5 Master List a read and see of any of that applies to your installation.
Let us know one way or the other. There are things you can try that don't really require you to be too geeky. ;)