Home owner insurance rates are partially credit score driven. Good credit = lower premiums, etc. Also, the state of Florida has sponsored a website where you can do some research. I've heard both good and bad about the site. Here's a link -->
Compare Homeowner's Insurance Rates.
Newer homes are built to higher wind standards and will have cheaper premiums as well, but most of Pinellas County was completely built up by the 1980s. Palm Harbor has the most new(er) homes in the county. Oldsmar crosses two counties, Pinellas and Hillsborough with the newer homes in Hillsborough County.
The FEMA flood designation will determine whether you need flood insurance, but it's not expensive.
You might get more responses if your post was in the Tampa Forum.
Hope this helps a bit!