Oh yes they do!!! Happened to my team last year. We had to reschedule our very first LI Cup game.
Wait - it is highly unlikely that an LIJSL team that was participating in the Long Island Cup would also have been accepted to the Jefferson Cup.
Are you telling us that your Long Island Cup game was rescheduled for a State Cup game that was moved due to a premier team playing at Jefferson? That seems the more likely scenario.