I moved from two 7D's to a 5D3 while waiting for my 1DX's to arrive (I wanted to shoot FF and also take advantage of the higher ISO capabilities of the 5D3 over the 7D...)... I shot 14 days of a little league tournament with the 5D3 (shot the last two days with the 1DXs)... don't get me wrong, I loved my 7Ds and had great successes with them... but I shoot a lot at night under the lights and in gyms that have questionable lighting... (none of which "allow" me to flash) and just as I sold my 1DMk2 the day I got my first 7D... I sold one of my 7Ds within days of first shooting the 5D3 (had to have a backup if the 5D3 died...). When the 1DXs came, off went the other 7D, I knew I would never use it again...
The main reasons...
Higher ISO capability
Better IQ
Better AF system (all around... points, speed, etc.)
larger screen (anyone here getting younger, do tell your secret!)
Because FF, beautiful DOF shooting wide open... (but that's probably one of your reasons for wanting FF...)
During the day, the 7D is great... once the light starts to get questionable, the 5D3 will step up and allow you the shots that others cannot get... this is a business for me, that is critical now with soccer mom and photo dad being my biggest competitors...
In your situation papa-razzi, I'd consider selling the 7D to help offset the cost of the 5D3... will you really use both? Once I've gotten used to the newer gear, I have a hard time going back... these days newer really is that much better...
I'm also having a similar dilema as you for my "family" camera... I have a 60D, which I had to put into business operation once when I sent off my 7Ds for service... I found the WB better and more consistent on the 60D than the 7D (it's newer it should be better right?)... but I simply LOATH Canon's 9 point AF system... yet now with two 1DXs and a 5D3 as "business" tools I'm finding it hard to look at anything "lesser" and not kinda cringe at the thought of not having the performance, responsiveness and capabilities of these two fine bodies...

The troubles we have in life! HA!