If you turn both volumes down, do you still get a sound?

If you haven't played a Jazz bass before (you may be used to a single pickup bass, say a Precision) you should find that there are three basic sounds. Neck pickup only, bridge pickup only and both pickups on full volume. To me there's never a lot of variation in the sound level or tone with say 100% neck and the bridge much less than 80% (or vice versa). Any variation in tone or output is very subtle indeed until the pickups are near equal volume.

So if you have one pickup volume on full and are adjusting the other pickup volume and expecting a wide sweep of tones, you are unlikely to get it.

If the volume pots are working you should get:

Both volumes off - no sound.

Neck volume right up, bridge volume right down - neck pickup.

Bridge volume right up, neck volume right down - bridge pickup.

Bridge and neck pickup both volumes up - mixed bridge and neck position.

Tap the pickup poles with a screwdriver to confirm which pickups are working.

With one pickup volume right off, you should hear the other pickup volume control working as expected.

If you aren't getting that, and its full volume from both pickups all the time, then there's definitely something wrong somewhere, but it can only really be the ground connections to the third lug on the volume pots. You wouldn't get a signal from a pickup if there's no ground connection from the back of the pot to the jack. It may be worth just reflowing those solder connections for each bent tab to the back of the pot with new solder.