I've been playing a Jazz Bass since 1966. Originally had a 1965 Jazz that I bought in '66 and was stolen along with all my touring gear in 1976. Was able to find a Jazz with the same '60's slim C-shape neck with 7.5" radius in 1985 and been playing that ever since.
My '82 Jazz has a maple neck and 70's era pickup spacing. It's a little brighter sounding than my '65 was and has better sustain. I credit the sustain to the all maple neck and the brightness to the 4" center to center spacing that puts the bridge pickup closer to the bridge than the 60's and also current pickup spacing, which is 3.6" center to center.
I run my Jazz with all three knobs (Volume, Volume, Tone) maxed. It's a passive Jazz and all of those came with what they call the Grease-pot tone and volume controls. Basically there is no such thing as boosting anything. turning down those controls cuts the sound that would otherwise come through. So I keep those controls dimed and adjust my EQ at the amp. Running those all dimed will give you really nice sound, but you will want to boost the low-mids in your amp a little bit. I run both DNA and Eden amps and I boost the low-mids to about 1:30 on both of those. If you turn them up too high, your sound will honk at you, which sounds bad. If you hear that, just turn your low-mids down slowly until it disappears.
My DNA-800 amp centers the low-mids at 500Hz and my Eden amps have semi-parametric controls that I like to set to ~250Hz. Anywhere in between those will work. I've also done some recording at a pro studio where I used a WTDI (Eden DI pedal with built-in WT preamp) for the 3-hour session. You can toggle that to either 500Hz or 2.2KHz. I set it on 500Hz and the raw recording came out amazingly good, to the point the owner and the sound engineer both came out to see exactly what I was using and how I had it set.
Don't expect to get your Jazz to sound like your P-bass. It can get close, but you need to use your P-bass to get that real P-bass sound. With the Jazz, leave the neck volume at full, lower the bridge volume to about half, and then slowly turn down the tone knob until you get close to your P-bass sound.
Those low mids are really important to being heard in a mix. Bass drum owns the lower frequencies and guitars and keys own the high-mids and highs. A little boost in the low mids gives definition and presence to the bass.
So, I'd suggest you go to your local Fender dealer and pull one off wall. Plug it into an amp; set the controls of the amp flat except for running a little boost in the low mids and see how she sounds. With my Eden amps, the bass EQ is a shelving control set at 30Hz. I always leave it flat. My DNA-800 amp has the bass EQ set at 35Hz and I boost the bass to about 1 o'clock to get the same sound out of my bass EQ that I get out of the Eden amps.