American Airlines Kate Hough reference 1-28016155904 Refused to issue replacement ticket and kept my $306 Ft Worth Texas

The fact that the plane changed gates has nothing to do with you not making the flight. The only reason you failed to make the flight is because you failed to arrive at the airport in time to be able to check-in before their cut-off time. For American you are required to check-in and drop off your bags 45 minutes before the scheduled departure. If you arrive at the agent even a minute late they have a right to deny you boarding with no refund or other compensation. This is the same for ANY airline unless you had purchased a fully refundable ticket(which for $306 I can guarantee you did not). However, I would bet that you were not just a minute late.

Had you checked-in on-time there would have been no need for you to "run" to the other gate, as you still would have had plenty of time to make it. I've been to DFW and it would take you perhaps 5 extra minutes at a brisk pace, at a normal pace probably 10-15 minutes. If you were airside, you could take the terminal tram and be there in about 5 minutes.

So by you saying you would have had to run, confirms that you did NOT arrive at the airport with sufficent time to Check-In and get to your gate. Which for  Domestic Flights it is recommended to arrive 2 hours before, and since this was a holiday period perhaps even a bit earlier. That is not driving into the airport, unloading your bags at the curb, or looking for a parking spot. That is with you physically inside of the terminal.