Can I legally begin a new job while on paid vacation?

I am about to be accepted to a higher paying job. The only things left for me to do is to sign paperwork. However, I still have three weeks of paid vacation time at my current company, plus three days of paid sick leave. If I took the vacation time, can I legally begin my new job and get paid from both sides? Then I'll come back to my current company to give a two weeks notice of resignation?

Edit: I live and work in Washington state, if that is necessary.

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