I will second the info about cutthroat of all varieties being in the river all of trout season, and it varies on each stream. I have caught some very nice fresh sea runs on a handful of coastal rivers in June and July, some even with sea lice on them. This is very accurate information.
Wet flies sure do work especially if you’re fishing those slower moving pools/ frog water areas.. but being the dirty bobber fisherman I am, I have caught a lot more of them with nymphs/dead drift. Egg sucking stoneflies or rubber legs, general attractors like ice dub prince, possie buggers with rubber legs. A lot of the fish especially in the middle of a hot summer during mid day much like steelhead will tuck up into bubble curtains at the head of those runs and they stay down and aren’t as readily catchable on a dry or light wet fly presentation.
My absolute favorite sea run cutthroat fishing is in October after it rains and they’re eating salmon eggs. Usually fish a little shrimpy/fleshy colored bead head bugger with a bead or yarn egg type dropper and catch a ton of fish that way. All of my largest sea runs have come during this time with this setup. You also do have a decent chance at running into a steelhead or salmon with this setup.