Is there a way to take care of the AZ state case in WA state?

Not in my experience. I was able to help resolve a similar problem with someone who had a warrant in another state, by getting them in touch with the Public Defender's Office in the County it was issued. If ti were me, I'd put some of the communication to them in writing so that it is clear you are trying to resolve this if it goes to warrant and you get stopped by law enforcement.? For a minor offense, they're less likely to move you to another state, for a greater charge, the odds go up. A lot is going to depend on the severity and legitimacy of the charges. There may be some flexibility with the local Courts and phone appearances, but if they are pursuing this, they will likely want you there at least once.

No, you will eventually have to come to AZ unless you 1) hire an attorney and 2) persuade a judge to let you do a telephonic plea (which will not happen with a felony).