I am a convicted felon in Florida (simple drug possesion) and have heard that if I move to California, it will not follow me because California does not recognize or communicate with Florida's legal system. I also heard New York is the same. Does anyone know anything about this?

Why would you think that there is NCIC? For just that reason... to track criminals in other states.

Convictions follow you everywhere. Recently my cousin was arrested in Virginia for warrant he had in New York.

All of the states use the NCIC system