Hi, ca, and welcome to this area! You will have SO MUCH less rain in Sequim than you've had in Anchorage! :-)
About a pediatrician: here are two terrific resources --
UCompare Health Care from About.com
Doctors in Sequim, WA (Washington Physicians)
This lists 4 pediatricians in Sequim who apparently specialize in adolescent medicine (who knew there was such a specialty??).
The Health Grades website is:
Pediatrics Washington (WA) - Find Local Pediatricians - Reports & Ratings
They indicate no pediatrician in Sequim (as of that database, anyway), and several in Pt. Angeles. Pt. Angeles is only a few minutes' drive from Sequim and because both towns are rather small, they sort of "share" some resources. If you were to decide to live at the west end of Sequim, you'd almost be in Pt. Angeles.
In any event, I've found the folks in Sequim to be friendly and helpful, and I'm sure as soon as you arrive you can start asking everyone you encounter, and they'll give you their best information.
In regard to children's activities, I'm sure someone knowledgeable will come on this thread to reply. Have a great move!!