This is a common misconception, but it's absolutely false.I thought they couldn't have them in NJ
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You can absolutely possess hp ammo in your home, to and from the range/hunting/fishing (firearms code 2c subsection F).
I was actually arrested in NJ on a charge of possession of HP ammo. I could prove I was in the process of going to a place to go hunting so charges were dropped with apologies from the court. You may not go to the grocery store and have a box of hps in your car though, or even have a dummy hp bullet keychain as it's the bullet that is illegal outside of the house, not loaded ammo. It is 100% legal to possess and own though. You just have to be careful about where you take it (but you do with fmj's for that matter).