I haven't played any G&Ls yet, so I can't comment on them.
I've played dozens of Jazz basses (Pre-CBS, MIM, Marcus Miller, Jaco Pastorius fretless, etc.) and they all have that basic, growly Jazz bass sound, which comes through slightly different with each model, but, you can still tell it's a Jazz.
Jazz basses are super versatile. You can get almost any tone you'd want from them, from that spitting, Jaco tone to an ALMOST Precision bass tone. They get you in the neighborhood, but, make you walk the last few blocks.

They all have that slim Jazz bass neck (unless it's a special order) which most people like.
Jazz basses are nothing short of being a great instrument.
Now, this is why I'd recommend a Stingray or Sterling.
Stingray: A very pleasing, fat, delicious tone that can cut through anything, a neck that just feels right in your hands, (for most people) and an EQ system that couldn't be a better match for its pickup. Don't let the fact that it has only one pickup lead you to think it's a "One trick pony," as some players say. Stingrays have a naturally aggressive tone, but, if you experiment with the EQ settings, and vary where you pick/finger the strings, you can get some nice mellow tones that don't get all mushed up in the mix.
You can get a Stingray to sound close to a Jazz, but, the reverse is not true. It's the Stingray's EQ.
Sterling: A tone very similar to a Stingray's, though I think it's not quite as chunky as a Stingray's. Not better or worse, just slightly different. But, you have the 3 way switch which makes it more versatile that a Stingray. Add a slimmer neck (very similar to a Jazz neck) and you have the Stingray's slightly leaner younger brother. You can't go wrong with either one. I have a 'Ray, but, I still want a Sterling.
In the end, you really have to try all of them for yourself. Only you can decide, and only you can prevent forest fires.
I'd suggest trying a Stingray first to establish the Musicman tone, then try a Sterling to see which tone and neck you like better.
Good luck.
Mike