Question on car property tax after moving please

I moved in the current state in April, before I moved I checked with
the assessor office about my car property tax, they told me I don't need to pay
because I moved out before 05/31/13, but they sent me a bill recently, said
I must register in new state before 06/01, then I don't need to pay the old state tax.
I did register in new state in june, not before 05/31 as they declared.

I don't feel it is fair, they didn't answer my question clearly even I consulted with them before i left.
now I need to pay two state's tax? If I deliquent on the old state' tax, what will happen to me? They will sue me?
Thanks for your information.