Booger's got it exactly right between the 2 you asked about - 3 Dives and DeBar.
How about trying them both different days and giving us your report? - everyone has different tastes.

Now that you've found several jerk chicken places, and gotten new suggestions, too, I want to suggest an expansion.
Try going up to Red Dragon for their jerk pork - it's for lunch, though, not dinner.
It's ready about 12:30 or so, and served until it runs out - so is why not dinner.
I also don't like my food hot (spicy) - and their jerk pork can be. But if you ask them to pick out some pork and leave out the hot peppers they can do it no problem. It will still have a likkle of the hot flavor, which I do like, but not burn you. And their Red Stripes are really cold - go perfectly with it.
Any taxi driver will know where it is.

For the ID off resort - the only time I carry it is when I need to cash travelers checks.
But like some suggested, I also carry a copy of my passport ID page all the time.
It won't let you cash travelers checks, but in case the worse happens - like something medical - they will know who you are.