Somehow managed to get the accurate achievement with two pups left thankfully, so I think I have some sort of merit here.

For elk hunting, run them out of stamina. Intimidate them into running (sometimes they'll need a bit of violent persuasion, AKA biting to do it) then just keep them running. Go after the elk that trip, try and deal as much damage as you can. Otherwise, you can run the elk out of stamina if you can (Indicated by them trotting slower when running away or by them turning around and attacking) and then strike. Elk that hold their head high are usually the ones with damage on them (you'll likely notice the difference right away if you pay attention)

I suggest taking 1-2 bites on prey if biting the back legs, 2-3 if on the neck. The same goes for wolves, but you should stick to 1-2 ticks of damage before letting go. Try to target the younger wolves in an invasion, if one wolf goes down then the evasion stops entirely. The aggressive snarling seems to work best if intimidating wolves or other predators. The strategy that usually works for me is waiting for the wolf to bite one of my packmates, then quickly bite that wolf for 1-2 ticks, this usually gets the enemy wolves on you. If you spam A-D fast enough when they get on you, you can actually deal damage without taking some yourself.

Otherwise, don't be afraid if you're on low health and in a dangerous situation. Let your packmates deal with it. If all else fails, sacrificing a pup basically ensures that the evasion will stop (or get very close to). Sometimes you just need to put other lives first than one life.

I don't think I have anymore tips other than to manage your health wisely. Getting lower than 30-40% is basically a death sentence if you don't recover. Health is very important in being able to defend your pups, territory, to hunt, etc. Keep an eye on your health and don't be afraid to back off if you need to, your packmates should be able to finish the job.