Re: 6.5 or 6.8?
The 6.5 will have more energy transfer at longer ranges vs. any other round with a similar weight. It has the highest BC of any round you can run through an AR. 6.8 was designed to have more knockdown power within the average range of the M4 --which is considered to be 300m.
The 6.5, and I graphed this myself against data I found and had from many different barrel lengths and loads, will retain more energy per grain than 7.62 with similar barrel lengths and "average" bullet weights, so 120gr. for 6.5 and 155-168 or so for 7.62. 7.62 starts out faster with more energy, but that changes. At about 600m, the 7.62 slows down to the point that it and the 6.5 are sort of equal in velocity (though 6.5 started out slower) and by 1000m, the 6.5 will generally be travelling faster than the 7.62 will. Also, as you can probably guess from the above, the drop is less severe in the 6.5 too.
For all practical purposes, the 6.5 does everything the 7.62 does at long range, but in a lighter package with a slightly smaller bullet. It is more accurate. At close range, it is significantly more powerful than 5.56, but not nearly as powerful as 7.62 up close.
For your ranges, yeah, I'd say you could use either of your rifles you have now. But I have a 6.5G and it is my favorite of all my rifles. .33MOA (mine best with handload 100gr. Hornady AMAX for some reason) good to 1000m (with a heavier bullet) light AR platform. It could very easily be a "do it all" rifle, deer, hogs, coyote, target, competition, 3-gun, defense, whatever. Energy transfer? It hits plates as hard or harder at 200m than the .300BLK firing 125gr. loads, similar to 7.62x39. Both hit much harder than 5.56 and will swing the hell out of a 12" plate on chains with one shot.
6.8, and I don't own one but I did look a lot into it, just didn't cut the mustard for an accurate weapon to 600m, much less 1000. It has a relatively poor BC, and that translates into significant loss of energy over distance, and a greater, accelerating drop at longer ranges. Basically, the 6.5 can do everything 6.8 can and more. It also has greater range in bullet weight than 6.8. At close range, they both do about the same thing when firing the same bullet weight. Oh, and 6.5 bullets are more common, they are used in other calibers, so for handloading that is a plus.
Now I mentioned .300BLK, you look into that? To 300m, it does great, much better than I had anticipated. Like I said, the 125gr. load is nearly equivalent to the x39. But unlike the x39, .300BLK probably has the greatest bullet weight range of any AR loading. It will essentially use every .30 bullet from around 100 grains to 220 grains, and some handle 240's okay too. With the 220's, it is very, very quiet. No ear pro at all needed, you can clap your hands louder. Now if you had that, you'd be able to put a suppressor on it and choose between subs or supers. For hunting hogs, you may want to keep with the supers, likely 125's, but you can also use the middle weight bullets, like the 147 to 155. The .300 I have shoots about 1MOA with good ammo (I haven't loaded any yet) and it outperformed my expectations in every way.
In a nutshell, if you want long range capability in an AR platform with laser beam performance at close range, the 6.5 is it (I honestly think it is the caliber that maxes out what the AR platform is capable of). If you only want 300m performance and good knockdown power (and the ability to suppress subsonic loads and supersonic loads along with all the various bullet types it can handle) then .300BLK is it.
Oh, and not sure about 6.8, but .300BLK and 6.5G both function well with short barrels, they don't lose a whole lot of energy like say, the 5.56.
Finally, I understand the ultimate hog gun is the .50Beowulf, but you have to get within 200m of them with lighter bullets, 100 or so with heavier ones. It knocks them down like a thunderbolt. At and after 200m with 334gr. bullets, they start to drop like a rock. I have one of these as well, and it is definitely a blast! But very limited in what you can do with it.
Hope that helps, I'm an AR nut and I love the high performance, relatively obscure cartridges.