My choice would be the 5D3. I upgraded from the 7D to the 5D3 and it's hard for me to go back to crop. Crop images lack the pop of FF images. FF has better color depth, better DOF control (as in smaller DOF options), better low light performance, and is most often sharper than crop. Many overlook the latter. The "reach" advantage of crop is offset a but by the fact that crop will give softer images than FF with the same lens. Take a look at the link below which compares the 7D2 with the 1Ds3.
http://www.the-digital-picture.com/Reviews/ISO-12233-Sample-Crops.aspx?Lens=972&Camera=963&Sample=0&FLI=0&API=0&LensComp=972&CameraComp=453&SampleComp=0&FLIComp=0&APIComp=0
I shoot a lot of events and sports (mostly indoor) and prefer to use two bodies -- one with a short zoom and one with a long zoom. For a while, I used the 7D as the second body to the 5D3 and it was frustrating. I wanted every image to be as sharp and as clean as possible. When I switched from a long zoom situation to a short zoom situation, I found that I was swapping lenses between the two bodies. It would have been easier to shoot with a single body.
The 5D3 AF system is extremely good, every bit as good as the 7D. I don't miss the extra 2 FPS of the 7D. However, I do have to be more selective when I burst in RAW because the 5D3 buffer fills up faster. Still, this hasn't been a problem. In low light, I have found tracking sometimes loses focus when the subject lacks contrast. But, playing with the Case settings has resolved much of this.
I'm in a situation where I occasionally challenge the limits of the 5D3 with low light sports -- such as with figure skating. 6 fps (and 8 fps of the 7D) is too slow to rely on. Skaters can rotate 180 degrees between frames. I have to rely on timing, instead. I've been tempted by the 7D2 with hopes that it will track better and the 10 fps may increase the odds of capturing a mid-jump shot with the skater looking in my direction. But, I can't go back to crop for these low light situations. The 7D2 might give me a few extra shots that I'm missing with the 5D3, but every shot will lose the extra pop that I get with the 5D3. My keeper rate with the 5D3 is quite high and I'm not willing to drop down in IQ for the sake of a few extra images.
I understand that the 5D3 still has at least a stop advantage over the 7D2 in noise. I live between ISO 3200 and 6400, so that extra stop is important to me. If you most often shoot below 3200, noise may not be a factor.
For action with more light, perhaps the 7D2 will do fine. It's worth noting that an f2.8 lens on crop has the DOF of an f4.4 lens on FF. So, crop does benefit from this extra wiggle room when tracking moving subjects. It can be off just a tad and subjects will still appear in focus. If the FF body loses focus slightly while tracking, it may be more noticeable. However, when it locks on, the FF body will still have more pop with the smaller DOF and the sharper images.
I must stress that the 5D3 has a fantastic AF system and is an underrated sports body. Personally, I'm in a position to up my game and seriously considered the 7D2 for its advantages with AF tracking and 10 FPS. But, for the reasons stated above, I'd rather stick with full frame. I've decided that the only path up for me is a 1Dx (which is on order, thanks to dropping prices).