It's counter-intuitive, but the state extension payments are entered in the same place that the federal extension payments are. You noted that "It asks if I want to request an extension" which tells me that you are looking for it in the state interview, but it's not there.
Have faith and follow DoninGA's instructions and look carefully - you will see the reference to the state payments eventually. When you see a reference to "Made any other federal, state or local payments in 2018", answer "Yes" and you will eventually see it.
