Apperently getting cold makes Elsa go to drunk-like state. 'And yeah, Annaäs and Elsa's parents were kinda bad at parenting. Well, former king of Arandelle is voiced by Maurice LaMarche who voiced Big Bob in Hey Arnold so he has exprience voicing crappy dads. Well when really think, episode Olga Gets Engaged is pretty much Frozen before Frozen.
If your cold is bad enough, it can tire you out. I've had colds like that before.
I find the idea of Elsa getting sick is a touch of Fridge Brilliance, no pun intended. She was so set on avoiding people she wasn't exposed to germs much. Now that she's more social, this is coming back to bite her.