Here are your criteria;
Is it the same airline?
If not, same terminal?
How many flights between DFW and your final destination?
Is this high season? Spring Break time? Or a busy time of day?
Two adults? (as asked before).

I personally, can kill ages of time in an airport with kids. They always seem to find stuff to do. I've had 6 hours' connection and once an 8 hour delay. They were wandering around, checking stuff out. They even find other kids to play with. By the time you change him, feed him, etc. Maybe a visit to the restroom or a quick bite to eat, head over to the other gate and presto! Departure time without the stress.

I personally, don't like to be stressed when flying with my kids, especially when they were little. But again, I was usually alone...

Can you get rid of that stroller? Since it's a domestic, probably daytime flight, how about using a good baby carrier and then put the car seat on either a folding metal luggage cart or a rollaboard (although I don't recommend the latter when traveling with children). Either check the stroller as luggage or borrow one at your destination. Obviously easier if you're visiting friends and/or family. If you decide on the 1 hour connection, do NOT gate check the stroller. That's a no-brainer.