To start over and pick one only? 6.8.
It would be fun to check out and it is a bit of a compromise between both cartridges. It isn't too big for anything and it can drop North American game easily.
The .308 is extremely popular and therefore extremely well tested and can be tailored for exactly what you want. However, yes, the trajectory is like a football! I still think it is my favorite though. I would only go 6.8 because I could have the compromise of shooting it out of a smaller AR-15 and being able to shoot rapidly.
The .556 is ok at best. I know some people love it, but even they say "it is enough of a round for what you need", well if it is enough, how about a little more?
Drawback of 6.8.... who knows when they are going to stop producing it? At least with .556 and .308, they are going to be made for a LONG time since they are both NATO rounds.