I don't know what I would do if I were you. I know everyone keeps saying that you can get sick ANY time of year. That is correct. But I've also walked in your shoes, so to speak and find that Jan-Apr was filled with constant illnesses for my kids. They are now 12 and 9, but my 9-year old (he has asthma), well it is just the pits. Any little cold will just get him down, despite being on all the preventive medications. And at age 3 and 5, it was CONSTANT. I, too, would be thinking twice about going on a trip in March. The stress was just too much for me back then.
I will say though that we did also have a beach trip in August a few years back. DS was about 5 at the time and he had a *cold*. Well, the cold turned into some miserable thing and he was miserable and sick. By the end of the week he had a fever. Everytime he went in the ocean he had snot draining out his nose and the other kids were making fun of him. Turns out he got a horrible sinus infection. The only saving grace for me is that the beach is only 3 hours from my house, I'm familiar with the urgent care clinics there and if I need to get home quickly, it is no big deal. Not so with Disney. It's really hard to spend all that money and have to deal with dragging a sick, asthmatic kid around.