The 5D classic is a pretty solid camera. I had one for a year and a half and just upgraded to the d700. I think it was a great camera, especially for a first time FF user.
It will make your 24-105 way better than it will ever be on a APS-C body(i hated my 24-105 on my 550d and loved it on my 5d.)
The camera is very basic, which i think was really cool as it was almost like using a film camera... center point AF, no fancy anything, simple everything... I just capped the ISO at 1600 and it was pretty decent. Nothing amazing for low light, but for playing around, and using for general fun, it was totally worth the 800 dollars I spent on it in 2010. I sold it for 650 and don't regret it for a second.
When I first saw your post i thought you should just buy the 5DII, but if you want other lenses, etc, just sell your T3i for $400-500 on craigslist, and pay 100 bucks to upgrade. Unless you really care about the video, i would say you don't really lose anything except maybe a couple hundred bucks. If it was possible, i'd try to get a used 5DII, but tbh, I never really liked the 5DII all that much for the price.