My primary bass used to be a late 80s G&L SB-1 (essentially a P bass) , and although I love the sound, the 44mm nut and the weight make it a difficult bass to play for 3 sets. I picked up a RW Jazz a few years ago, and immediately felt the same as you described. I bought a set of DiMarzios, and a heavier bridge, and although it improved the sound, it wasn't as good as a P bass in a band setting. The RW didn't really work for me until I started from scratch with my EQ and started using either flats or half-round strings (I like Ken Smith strings). I recently added a Dark Glass Vintage Ultra to add just a bit of growl, and now really like my sound in a band setting.
I'll make the same statement everyone else has made though - a Jazz bass is not a P-Bass, and you are never going to have that sound unless you maybe throw some Precision pickups in a Jazz bass. If you come from playing a P-bass to a J-bass, it is a very unsettling change. I played a Rick for years, and I had a similar difficulty adjusting to the sound of my SB-1 at first. So what I'm trying to say is you have to just decide if the bass sounds good in the mix as a bass, rather than comparing the sounds from where you stand to that of a P-bass. The front mix might be better than you think. For sure it will be different.