UPDATE:

I have overnighted at DFW Terminal A and would be heading to the Admirals Club when it opens in 15 minutes.
I had no issues staying overnight airside and I was not the only passenger that had an overnight connection from our flight.
Maybe I counted 10-15 other passengers at the airport that were overnighting and resting at empty gate areas.

The only thing open overnight is the 7-Eleven stores. I didn't go landside but some of the security checkpoints at the airport are 7/24 open so if you mistakenly go landside you could go back, although they are only in Terminal C and E but you could take the Skyline between terminals or walk within a reasonable time.

It was definitely not worth paying for a hotel for the full nightly rate for a half-night.
I don't have PriorityPass, but for those who have a PP membership, the Minute Suites at Terminal A were open.