I made a simple mistake on my federal return with e-filed already , over-reporting on interests (from what are on my 1099-INT forms). So I will need to do a 1040X later to get a refund (over $300). Since I plan to file the State Return with paper filing, should it be easier if I just correct my interest income on my state return and get it done correctly. Would the State know there is difference between my 1040 and on the State Return ?
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).