Yes. When you amend your federal return, any new information will transfer over to your state return (if it changes state). If you see anything besides a $0 in the state refund/balance due box, it means your state return has changed.
TurboTax will update your state numbers, but you still have to complete the state amended section and print and mail your state amended return if you want to amend your state taxes.
You definitely should file a state amended return if TurboTax shows a payment due. If you are getting an additional state refund, it's up to you whether to amend state or not. If you don't, you'd be leaving money on the table.