Hey Kat, I've never done it, but I know plenty of employees who have done sleepovers for school/Scout groups (I imagine family sleepovers would be similar). Usually they take a tour, maybe have a quick class or Q&A, eat some pizza, then spend the night on the floor *of one of the animal exhibits. I think Shark Encounter and Wild Arctic are the most common locations, I've heard of them in the manatee exhibit as well, probably elsewhere.
The educators stay with you all night, and you aren't allowed to leave--even just to use the restroom--unescorted. You wake up and go to breakfast early, before the park opens obviously.
Personally I'd stay in the All-Stars before I'd spend a night on one of those nasty floors in a sleeping bag, but then again, it's been decades since I was an adolescent boy. Your sons might find it more appealing.
*-Some of the educators bring cots. I don't know if grown-up participants are allowed to, you might want to ask if you decide to do this.