You need to file a part year resident KY return. You will report all your income received, after the move, regardless of where earned.
You need to ask your employer to start withholding KY tax. If he is unable or unwilling to do so, you should consider making estimated tax payments to KY.
TN does not have a regular income tax. Tennessee does have a state income tax on interest and dividend income from investments. You only need to file a TN return if you have taxable interest and dividend income in excess of $1,250/person ($2,500 for married filing jointly).
