I'm not to sure about Jax Beach but Vilano Beach in St. Aug is good surf fishing. I used to go every Sunday to relax before I started my CJ7 Frame up build. I'd like to go soon though so if you would like to go PM me and I'll give you my number.
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I'm Nicks little brother, surf fishing is pretty good a Vilano as I've gone with him several times. I remember when we were down in Juno Beach in the Palm Beach area in December it got quite cold here in Jacksonville and I guess the Pompano ran south because we were catching a ton of them. Pompano is our favorite, we catch pomps at Vilano from time to time. Reds, whiting, sharks, flounder and unfortunately sea cats are what we normally catch at Vilano with the occasional Pompano.
Huguenot Park off hecksher drive is a good spot to surf fish.
I used to go fishing there ALL the time with my dad and always caught something.
Beware of the jellyfish though. You are allowed to drive on the beach there and its something like $5 to get in.