false.. you can gatecheck your carseat too. i prefer this as it minimizes the handling and increases the chances it DOESN'T get damaged..
southwest's Contract of Carriage (page 25) states:
Travel equipment for infants or small children Two (2) items of travel equipment for infants and small children, such as walkers, child safety seats or collapsible playpens, may be checked and transported in substitution of one piece of the free baggage allowance for each fare-paying passenger at no charge.
So this means, since you get 2 free bags per Southwest policy, you can bring ONE bag, and then TWO child items (i.e. 1 carseat, and one stroller)
I'd gatecheck.. i've heard horror stories of damage when you check it at the ticket counter.
what we did was take the double stroller, load all the carseats (2) on it, put carryons, etc too. then wheeled the stroller around. at the gate, let them know you will gatecheck at the plane, and wheel your stroller monstrosity down the ramp. At the end right before you get on the plane, take the carseats and everything off, and give what you are gatechecking to the rep standing there.
We gatechecked the stroller and a carseat. they were waiting for us when we deboarded both times... We had a bag for the cehcked carseat too.
We flew Jetblue so JB doesn't charge for any of those and allows unlimited "assistive" devices..
I might add, SW states that if you bring a carseat for them to sit in, it doesn't count as the 1 carryon + 1 personal item.. so you can bring a carryon, a personal item and the carseat per person