On my 1099-R box 13 is the State/Payer's state #. On Turbo Tax box 13 asks for a Date of Payment. What do I enter into Turbo Tax for box 13?

Yes, "date of payment" would be nonapplicable (or more so, unimportant) to the IRS as they look at the whole year and not a single date. Just leave this blank and since you are in the 2020 tax program, the IRS knows it pertains to 2020.

.......OK, here's your situation with that 1099-R.

Whoever issued that 1099-R did NOT update their form to the new/proper 2020 1099-R box numbering format that the IRS wants everyone to use this year.   So your boxes 12,13,14 on your paper 1099-R will go into boxes 14,15,16 in the software's 1099-R form.

of course, if there was no withholding in box 12 of your paper 1099-R, then you may not need  to enter boxes 14,15,16 in the software's form at all.