There are definitely differences from state to state regarding state tax obligations, and you need to understand what NJ considers "ties". Generally, you will need to prove that you have broken ties to your former state of residence, and do not intend to return. Owning a home will likely make this a bit difficult.
With respect to your driver license, again, check the state requirements. Many require you to report a change of address, and some (VA is one) specifically state that if you move out of state or to a foreign country, you cannot legally hold a DL. If you have an accident, and your DL is considered invalid, insurance won't pay, and you could also be charged with driving without valid insurance. So yes, it's complicated.
Assuming you have income now and/or expect to have income from employment in the UK, you will likely want to avoid double taxation by opting into the tax treaty. You can exclude earned income or take a federal tax credit, but you need to meet the foreign residency requirements (either bona fide residence test or physical residence test). Accordingly, you need to consider how much time you expect to spend in the US very carefully.
