The sun does beat up off the walkways quite a bit ( doesn't it EVERYWHERE in Florida

) but unlike AK the park is pretty open so any breeze is felt by the guests in the park. It also doesn't get anywhere near as humid as AK seems to get.
There are a lot of indoor and cooler things to do at SW. I would try to schedule so you do some/all of the following things during the hottest part of the day Wild Arctic, the penguins, sharks, manatees and pets ahoy show. All are indoors and airconditioned. For the outdoor shows (Shamu, whale and dolphin show plus Clyde and Seymore show) I would suggest you try to get to the shows a little earlier than you would otherwise have done to make sure you get a seat in the shade. If you get to the show 20-25 minutes ahead of the start time that should be plenty. You can always use the time to have a soda or an icecream while you wait. They put "pre show" entertainment on for all the shows that last 5 minutes or so.
It would be almost certain they will run a second day free, so I would advise take advantage of that offer to make sure you can do the park at a pace that suits your family. They have a nice play area at Shamu's happy harbour and while the activity may heat up your kids a bit there is a water area filled with showers and fountains that would help cool them down. There are bathrooms in that area so if you wanted to take them a swimsuit and/or some thin shorts you can change them easily enough so they can have fun getting wet/colled off and then change back again afterwards into some drier clothing. The JTA ride is a great way to cool off, but you WILL GET VERY WET on it, particularly in the front of the coaster.
As gw says, the show times make it almost impossible to see all the shows in one visit to SW (without huge amounts of planning and back tracking) taking a second (even half day) to cover the bits you miss really helps to make the most of this park.
P.S. take plenty of money for fish LOL, at the ages your kids are there is a good chance they will want to be feeding sealions, dolphins and stingrays