For East Texas, 7mm-08 would be just what you need. But for the Mulies, in West Texas, New Mexico, or Arizonia, I'd go with something with a bit more range. I can't count the number of Mulies that I had to pass up because the gun I was carrying did not have the range needed for the monster buck across the meadow.
My brothers all laughed at me when I brought my 30-06 Handi to Tennessee to hunt Whitetail with them. They carry 30-30s, 444Marlins, and 45-70s. But they quite laughing when I dropped a nice buck across the hayfield at 400+ yards. As far as recoil goes, the 06 can be down loaded to match a 7mm08 in that department. From my experience a 30-06 is the bear minium for Moose. I've seen a lot of Moose lost with smaller calibers. Elk too. Yes I keep reading where people talk about Julie and Mikki Collins shooting their Moose every year with a .308. But what they don't realise is the patience and experience these two women have. They grew up here in the Interior of Alaska. they have the experience and know how needed to stalk close enough to do the job right with a .308. Few men that I know could do the same things they do. They are two of the most self sufficient persons I have ever met.