Yes, you are still eligible. Your WA employer paid into the WA coffers. And your weekly benefit amount won't change, as it's based on your earnings here for your qualifying claim year.
I was laid off in WA, and almost immediately moved to San Francisco. All I had to do was fill out the change of address online. The only stipulation was that I 'check in' with my local unemployment office in my new city once I was there. I kept filing my WA weekly claims online and my direct deposits kept coming. Every once in a while I'd get one of those random "we looked at your job search information" letters that basically said - looks good, continue as you are.
I just moved back to Seattle (obviously there are no jobs in San Francisco), filled out the same address change form and things have really been seamless. Don't be scared, if you think relocating will help, do it! Washington Employment Security really doesn't care.