You left out mentioning if you worked in KY or TN after you moved there in 2019. Did you work in KY or TN from Sept to end of 2019? If so, then you need to file a return for a partial year of residence in KY, If you did not work in KY after moving there, then you have no income sourced in KY to file a KY return for. And TN has no state income tax. If you have been working in KY in 2020, then you will prepare and file a KY return for 2020 when the 2020 tax season begins in early 2021.
We are missing some information from you. When you moved to KY, did you continue to work somewhere? Did you work at the old job in TN but live in KY? What did you do in 2020---have you lived/worked in KY or in TN?
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).