Thanks for all of the info!

Ended up getting into DFW about 2 hours late. My connecting flight was also delayed, and for a while it looked like I'd make it.

But as luck had it, I saw that plane being tugged out while my arriving plane parked just a few gates over.

When I saw that, I called, got a great agent, who put me on the next DFW-SLC flight.
The customer service area at DFW A was completely insane.
Walked to the gate where I saw the GA turning away some others, said flight full, go to next.
I approached, said, "I called and they said to come here for a boarding pass, and you may have an exit row seat?"
The GA answered, "yes, and there is an exit window. is that ok? it makes things easier when you call."

It sounds like I may have gotten the last seat on that flight.