PA and NJ do have a reciprocity agreement. This means you can pay taxes in your home state of PA and not have to pay NJ. The PA tax rate is about 1/2 of NJ.
Social Security and Medicare tax will not be affected by any forms that you changed, but you may have inadvertently entered something to increase your Federal withholdings.
It may not be a bad idea to submit a W-4. You will have to decide if you want a bigger refund, but I strongly recommend that you don't plan to owe on your Federal taxes.
With the reciprocal agreement, you should only be paying taxes to the state that you live in. You should not be paying to NJ at all.
Your HR department can refuse to take a form that is knowingly incorrect, but if you want to make a legitimate change you should be able to.
The state of your residence will always tax all of your income. Since you had been paying into NJ you should have gotten a credit on your PA taxes. Since the NJ tax rate is 6.25% and PA's is 3.07% you should not have owed PA taxes. Go back over your tax return and be sure you got the credit for taxes paid in another state.
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