I'm shooting the 65g Sierra BT SP's in my 223 with a 1 in 12 twist, and they're shooting great at around 2900 fps(muzzle), so those Hornady 68g bullets should be just fine in your barrel unless the velocity is just terribly slow.
Try Varget, and Rel-7, and the powders PaulS suggested. I'd say you'll be OK if you can get the velocity in the right range.
A bullet stabilizing really isn't %100 dependant on twist rate. I know that's what the common thoughts are on it, but it has as much or more actually to do with velocity. Bullets stabilize at a certain RPM range. This can be acheived with a fast twist at lower velocities, or a slower twist at higher velocities. That's a very simplified way of putting it, but that's the basics of how it works.