If I cancel my car insurance policy, do I have to pay the upcoming bill?

So, long story short GainsCo sucks and I'm moving from Florida to Texas. I've already got a new policy from Allstate. I went in to cancel my policy with GainsCo and they are telling me that I have to pay my upcoming bill (though it *should* be pro-rated) even though I am cancelling my policy. I've... show more So, long story short GainsCo sucks and I'm moving from Florida to Texas. I've already got a new policy from Allstate. I went in to cancel my policy with GainsCo and they are telling me that I have to pay my upcoming bill (though it *should* be pro-rated) even though I am cancelling my policy. I've already received the bill but that bill does not pay for existing coverage, that bill pays for coverage of the next period and I do not want to be insured by GainsCo any longer.

If I choose to not pay GainsCo for this upcoming period (for which they told me will be pro-rated back to me) will I receive any sort of penalties or anything like that?
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