I wouldn't have even ran after that first flight landed late at Dallas (DFW). The delay caused the misconnecton.

I would have walk normally / casually to the gate / re-accommodation center, request a Military Excuse (proof of delay) along with the reroute/rebooking, then book myself in a nice hotel, have a decent dinner at the hotel restaurant. Then file a claim with Citi when I get home and may be write AA to ask for some customer service recovery gesture mileage :-)

The rebooking might have even been at a higher fare class since it's close in. Sweet extra Avios!

And then the delay at Houston IAH? I would again go have a nice meal at a restaurant and file a claim later on AND since it's British Airways, a EU carrier, file a claim EU261 for cash compensation on top!