Ah, a first elk hunt. Nice to see through someone's eyes again what that is like. Have a great time! Hunting elk is very different from hunting eastern white-tailed deer, but in some ways can also be similar. One difference is when you blow elk out of an area they are g.o.n.e., no circling back or showing up in the same place tomorrow. They cover huge swaths of ground. But it is still possible to follow up spooked elk and get a second chance - as long as you don't let them know you are there. The lead cows are very crafty though.
Short answer, I would go .308. It will deliver a result very similar to the .30-06. I don't have one, and have only ever shot a deer with one, so I cannot be accused of being a fanatic on that caliber. But the rifle I did use was used by a friend for years on all his elk until he "learned that caliber was considered inadequate on elk". Hmm fed his family for years, and he eventually went back to it for his elk hunting.
My first few elk fell to a .270 Win. I was careful of shot placement, used Partitions, and collected meat. At some point, hunting Montana around grizzlies, I shifted up to 7mm Rem mag. I use that or a .338 Win mag depending on bear density. I am still careful of shot placement, but the larger two break bone better if I am off a little or the animal moves. Plenty of comments on calibers, but I got that you have the two and are not really looking to buy a new rifle just for this trip.
In CO my shots probably ranged an average of less than 70 yards, but from 13 to maybe 130. There I hunted dark timber almost always. In WY, there have been some long and some short shots, probably the longest closer to 250. Different cover of thick timber and larger park openings (grassy clearings), or ridge to ridge shots in those WY areas.
DRT does not happen except with certain damage. In my experience (from necropsying animals to evaluate damage) a spine or forward central nervous system hit is DRT. Hitting the major arteries as they exit the top of the heart is DRT. Most others run or walk a bit. One through and through double lung went at least a quarter mile at a gallop. Tricky tracking on dry pine needles, that one.
Personally, because of bad hits and horrible results I have seen on still living but doomed animals that others did not recover, I don't take head shots. Shot off jaws, shot-through swollen tongues, and other such damage does kill the animal eventually, and very cruelly. They can run a long ways, but they can't drink or eat. Like some commented on neck shots, you can shoot through a lot that is not immediately fatal, but they do die in a few days.
Just a few thoughts based on experience.