I did Discovery Cove and Sea World in 2009. I was in fact by myself that day, and it was July, so it was hot and the Sea world park was open later than I believe it is other times of the year.
I arrived at DC at the recommended time but was still put at a 1:00 pm time slot(which ended up being perfect).
I had breakfast then spent all morning snorkeling, feeding birds, rays, etc. I went around the "lazy" river a few times, I had lunch, then it was time for my dolphin swim. After the dolphin swim I went around the "lazy" river area about one more time, and back in with the aviary, snorkeling, etc...and was back to the locker area probably by about 3:ish? I'd say by 3:30 I was showered, and bought my photos, hopped on the shuttle and headed over to Sea World.
I saw the Clyde and Seamore(I think that's the name of it) show and the Shamu show I think it was the "Believe" show. I was also able to see most of the animal exhibits. I wasn't able to see Blue Horizons, or go on any rides(I get motion sickness anyway).......I was ready to see the night time "rocking" shamu show(forget the names it's been a while) but that's when it finally started to rain for the night. So I left for the day probably around 9:00 pm.
One of the most relaxing and fun days I've had.
Edit - Oh and if memory serves me correctly, the shuttle came pretty quickly and took me over to Sea World pretty quickly, so it's totally doable.