I'd say it depends on what you mean by "handicap". If the person needs to preboard for that reason, just tell the gate agent that he needs to preboard for health reasons; they will allow it. You'll probably get to preboard anyway since you have other children under the age limit. If the disabled person will need a boarding chair, then I suggest you call and have it noted, then ask about it again when you arrive at the airport, just in case. They don't always have one at every gate as a matter of course.
I'm wondering why you spoke of having two lap children, but said you bought six tickets ... who else will be with you? If it is another adult that person probably will not be able to preboard with your party. As others have noted, the 2 yo will need to have a paid-for seat; SWA will ask for proof of age on a lap child who is close to 2.
Is the flight totally sold out, or do you just mean the discount fares are sold out? If there are ANY fares left, you should go ahead and purchase the extra ticket ASAP; if you wait until you get to the airport you will pay even more, and run the risk of being denied boarding altogether. If it turns out that you see the fare for the flight go down after you purchase, you can ask for a credit on the difference.