I'm far from knowledgable about reggae, but I don't know of a problem with the Jazz bass.
I've been trying to learn some ska/ reggae stuff of late, & I've been using my Squier Jazz with no problems - as Fleabag says - neck PUP on, bridge PUP half, highs & middle off on amp, bass on full. I anchor thumb on neck PUP & play with fingers without digging in.
Sounds okay to me - just need to learn the right notes now!
There are people on here who will know about this stuff (like Reggaebass), who will be able to give you the 'low-down'
Maybe its not the type of bass - maybe you've just got freakish hands? Did you ever think that might be the issue?