So i saw that you make 57K a year and unfortunately (unless you get a roomate) that puts most of downtown JC (esp close to the light rail and path) out of your price range. Rutherford and Secaucus are similar to Palisades Park (nothing to do and you need a car to go grocery shopping). JC heights isn't bad but rent has been going up and only type of public transportation available in that area are buses. JSQ seems to be the most logical choice for you. Rent is not as high as other areas (but will get higher as they build it up). The area isn't a warzone so a single female should be ok but i wouldn't walk around by yourself at 2 am (same goes for downtown JC). You can do this in Secaucus, Rutherford, and Palisades Park but like i mentioned before there is really nothing in those areas.

Have you looked into Bayonne? Its more residential but its right next to JC, public transportation is pretty good, and it has a good amount of gyms and supermarkets. You can get away with not having a car in Bayonne.