Most of the teams around town have people who race cross and there are a few that are solely dirt oriented.
That said most teams are a bit more road oriented and really cross is an individual sport (though there is plenty of camaraderie - people put up tents warm up together, do a lot of socializing) Most teams recruit and bring on new members after cross season has started. We've been grappling with this a bit. We start recruiting in September/October and usually have a final roster sometime near the end of November. That leaves out a good deal of cross season. We're thinking this year we may have a separate cross recruitment a little earlier (though we still have a problem of not having kits.... we can really only do one order per/year, so scaring up extra clothing is sometimes hard). But anyhoo, where I'm going is that it may be a little harder to join a team for this cross season, but there's a good chance you could get on one part way in.
That's not necessarily a bad thing - cross is a pretty congenial crowd. You can go and hang out with people from all sorts of teams and get a feel for how they operate.
For learning skills there are a few options. Cycle U runs some cross boot camps. Those I've heard a pretty good. For skills and to meet people I'd suggest going to the Kore Kross workouts at Marymoor this fall. People of all levels go and participate and experienced riders lead exercises. You will not only learn a lot, you'll meet riders from lots of teams (and this is one spot where teams will be looking for new recruits)
A couple teams that I know are more dirt oriented are Ragnarok and 20/20 Cycles. I'm not sure how/when they recruit, both I could probably find out. If you want to just browse local teams you can check out the WSBA web site www.wsbaracing.org on the Links page there is a list of many Washington teams.