UF overcrowded?

My daughter is also choosing between UW-Madison and UF (admitted to both as an OOS student) so I found this discussion interesting. We loved our tour of UW-Madison and will tour UF for the first time next week. Unfortunately, we did not realize how quickly the dorms fill up at UF, and did not submit our housing application until yesterday. Is there a chance she will not get a dorm room since we applied so late? We hate to pick a school based on housing, but we’re learning it’s a bit of an issue at UF.