Moved to NH from NJ for work can I claim NH state tax?

Hi there

 

So I recently started work in New Hampshire and move up here from New Jersey. I am now living in NH but still have a NJ drivers license. I on my W2 it now has my NH address so I am not getting charged state tax. Is it that simple? Or am I missing something that I would need to do to prove to the IRS that I am now living in NH. Not sure if I need to do anymore.

 

Thanks!