Personally, I would drive because once you wait for the plane, wait on the runway, wait for luggage, etc. you could be there. We drive our kids to SA a lot and usually either leave early in the morning while they are still asleep or leave at naptime. It's only 4 hours from FW. Anyway - it's difficult to get around SA without a car. For Seaworld, I would recommend Radisson Hill Country - it's literally across the highway from the airport and has a great "beach entry" pool that's perfect for little ones. Plus, they run a non-spot shuttle to and from the front gate of Sea World all day long, so you can leave during the day and go back for a nap really easy.
As for flying with the big stroller, I have done it many times, you will probably have to fold it for the attendants as no one ever can. However, if you decide to go downtown to the Riverwalk on a weekend night, I would leave the big stroller in your room and take a backpack, if your child will sit in one. The riverwalk is handicapped accessible, but it's really really bad and there are stairs everywhere, which make using the big stroller a nightmare. It's also really busy at night, so there isn't much room and people will be accidentally hitting the stroller a lot. We've always used a hiking backpack down there and it works great.
Good luck and have a great time!