Self-parking is close enough that it (given all the walking you'll be doing anyway!) will seem like no big deal. If you're not going to be using your car, self-parking might make more sense. You can leave your luggage at the bell desk while you go back to park your car, so you won't have to lug it from the parking lot.
If you'll be using your car during your stay, it might be handy to have the car valet-parked, so they can bring it around for you. But, then, you've got to tip the guy who parks it and the one who brings it, so that can add up.
I believe the answer to your question about whether you would get charged both the resort fee and valet fee if you choose valet parking, is yes.