No, you wouldn't have to pay again, the $20 is good for the day anywhere you valet. However, I'd recommend charging the $20 to your room and not paying cash. I almost always get charged a second time at the second resort, and have to have the front desk remove those charges. That's harder to deal with if you pay cash. A couple times (before I learned my lesson) I paid cash to valet at the first resort, and still had an additional $20 charged to my room. WDW just took my word for it and removed those charges, but it's easier to dispute a room charge. I think the computer system they use for valet isn't quite up to the challenge.
