I've lived in St Thomas for 5 years and several in Puerto Rico too and am also a captain who's worked in the yachting and charter industry there.
Myself and quite a few friends stayed at the Crystal Palace (and Villa Fairview) in St Thomas. The two places are owned by the same guy and specialize in short term visitors. Google "crystal palace st thomas" and you'll find the site.
You can get news and want ads and everything else from USVI newspaper. Google "VI Daily News" and you'll find the site.
You should be in time for high-season which peaks in Dec but is just ramping up now. This means jobs will be easier to find but apartments will be getting more scarce as it gets deeper into season. Even in high-season apartments will not be impossible to find don't worry. Other than 'the projects' there aren't many apartment buildings in the USVI (maybe 10-20 between St Thomas - St John) and the ones there are are very small complexes not like what you'd expect from the states. The majority of apartments will be an add-on to someones house which they will rent out through word-of-mouth, networking, bulletin boards at the bar, etc.
Anyways good luck and have a kick ass trip..