Some blatantly obvious suggestions coming up, and I don't mean to offend, just trying to help:
1.) You do have the Bridge pickup doing the majority of the work right? Jazz basses sound great with both pickups on full, but if you want that growl you need to bias the pickup selectors to the bridge pickup. (obviously)!
2.) What amp are you using? Some amps that I have used don't have a large dynamic range that accommodate the large dynamics of a Jazz bass. Some amps I have used sounds cool for Precision basses, but I can't find a good tone for Jazzes, so that may be a factor.
3.) As far as strings go, I would say DR Hi Beams, or Sunbeams. I personally have Hi Beams on my bass, and didn't find the Sunbeams worked particularly well with mine, but nickel is supposed to be more magnetic and therefore make the bass sound more responsive. (I have a set of the afore-mentioned Fat Beams on the way to try). Rotosounds - Cheap and nasty in my opinion, but give them a go if the DRs aren't your thing.
4.) Are you plucking the strings by the bridge?
I hope something in there is helpful. I know it's obvious stuff but thought it may be of some use...
James