My 2p for what it's worth.
Ft Pierce is not a dive but it did when I was there have a bad area, specifically that around the infamous Avenue D. Vero Beach didn't really have a bad area as it was inhabited primarily by geriatrics.
Downtown Ft Pierce and South Hutchinson Island were certainly more pleasant surroundings than anything Vero had to offer. I can't actually think of anywhere 'close' to the airport where you are likely to find the type of shared accomodation you are referring to. Wherever you end up you are likely to need your own or shared transport.
Do none of the schools out there have apartments they rent to students anymore? If not, can't they provide accomodation lists, ie people they know that could put you up. I would be wary of just going out looking for digs unless they had already been recommended.
If you're going to be there for more than a couple of weeks I would consider staying in a hotel to be unsatisfactory. The option of eating out everynight soon becomes expensive whatever anybody might tell you. You might be lucky and get limited cooking facilities i.e. you could live off microwave meals. I have done a lot of flying in Florida over the years and as a student was always able to live in a house or apartment rented from the school or somebody connected with the school, and shared only with other students. I would have considered that the only way to do it and see no reason to view it differently now.
Come on guys, look after your customers.