You get your seats when you put your tushes in them. There are presently no seat assignments on SWA. You board with your lettered group, and you sit in any open seat except those in the exit rows. If you get an A boarding pass (even one A, in your case, since they won't make your kids board without you) you are assured of finding 3 seats together. Even with a B you probably would, but with B's it is worth standing in line at the gate to make sure of it.
You can check in at the resort by use a web-enabled cell phone, a laptop, or by taking your reservation number to the Guest Services desk at your resort or a park and asking the CM to do it for you. It is OK if you cannot print the BPs when you check in online, you can "reprint" them at an airport kiosk.
Curbside checking varies by airport; some airports have it, some don't.