How do I enter the amount I paid with state extension?

To enter, change or delete a payment made with an extension request (Federal, State, Local) -

• Click on Federal Taxes (Personal using Home and Business)
  Click on Deductions and Credits
  Click on I'll choose what I work on (if shown)
  Scroll down to Estimates and Other Taxes Paid
  On Other Income Taxes, click on the start or update button

On the next screen select the type of extension payment made and click on the start or update button

Or enter federal extension payment in the Search box located in the upper right of the program screen. Click on Jump to federal extension payment

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.