I work for a consulting firm in Georgia, and that is where my paychecks are issued. I am consulting for a NJ company, and I live in NY. Do I have to pay NJ taxes.

If you physically work in NJ - yes, you have to file a NJ non-resident tax return as well as a NY resident return. If you did not actually work or live in GA you have no tax obligation to the state of GA, even though the company is headquartered there. So your company ought to be withholding NJ income tax from your wages and not any GA tax at all.