You should not file a return that you know is incorrect (it would be a criminal offense as you have to sign your return stating that everything is accurate to your knowledge). So, you should file the state return with the correct interest information.
States generally have their own information matching systems but some use federal AGI as their starting point, so they can and do share information. However, this isn't something that would happen right away, so as long as you amend your return with the IRS your return would be updated by the time any matching occurred. And, if it were to generate any sort of notice for the discrepancy, it would be an easy fix - you could just respond with a copy of your Form 1040X (so just make sure to keep copies).