I live in Fairbanks but I have lived in Anchorage. Nearly half the population of Alaska lives in Anchorage so there's plenty of hunting competition around there. There's a few drawing moose hunts on the outskirts of Anchorage but you have to get further away to get into harvest tag areas. Yes you can get a harvest tag for moose every year but there are many areas that also require a drawing permit. Look through the regs and you'll see what opportunities are out there. Anchorage is in Unit 14.
http://www.adfg.alaska.gov/index.cfm?adfg=hunting.main
Jeff