Given your time frame from Anchorage, I am not sure how easy it will be to ditch the crowds, though it is relative of course. My own experiences have all been on the Kenai Peninsula, and there are some opportunities here and there for a little solitude but even the less crowded locales can get busy depending on the day. If you’re ok hiking a bit you will leave most people behind, but you need to be prepared to meet a bear or a moose. I have not ventured other directions from Anchorage, so it quite possible there are some less crowded options, but most of the promising streams I checked into have reputations for being pretty popular...
You definitely don’t need to join in a “combat” line up, but be prepared to have company of all sorts (delightful and obnoxious). It’s pretty common to find a nice hole for yourself only to have a few more folks pile in on top you. Often times I will go out of my way to actually encourage a friendly angler or two join me, as a good hole with “only”1-2 anglers fishing it is often a magnet for the kinds of A-holes you really don’t want to fish around. I have had to endure some pretty awful examples of just how ugly some humans can be, and have opted to leave a productive spot more than once. Have called the Warden in several of these instances, as snagging is a popular technique with many tourists and some locals as well. Etc etc... All that said, there is fantastic fishing to be had and I have also had the great pleasure to fish with some wonderful people each and every year I go. Just go with an open mind, and be ready to deal with some company.
P.S. I might end up sharing a hole or two with you, I’ll promise not to low hole you! PM sent