Perfectly safe to stay on the coast. You’ll be able to hike Mt. Si & Rattlesnake with waterproof boots— you won’t want to do both because they are quite similar. You can visit Snoqualmie— check out REI for gear rental if you don’t bring warm enough / waterproof items OR if you want to snowshoe or Nordic ski.
*Keep in mind these are essentially 3 hour drives without traffic.

However, you can easily spend your time in Port Angeles exploring the surrounding area. Which, IMO, is really lovely. You could make the 30 minute drive out to Lake Crescent or the 2 hour drive to Cape Flattery (the northwestern most point of the contiguous US & where the straight of Juan de Fuca meets the Pacific Ocean).

Enjoy!! Bring warm coats & waterproof outer layers! (And gloves and hats!)

Is the hike really worth it when you can still see civilization? IMO, no. It isn’t.

Is a hike really worth it when you also see an industrial complex (of sorts)? Nah. #1 reason I don’t hike Issaquah.