Looking for an Island getaway in the continental USA

Im looking to plan a vacation on an Island  for at least one week.  Id
like a variety of things to do including some quiet time, sightseeing,
etc...  Im not looking for something super touristy . Ive been to Hilton
Head Island, S.C.  and id consider that a bit too touristy for me.,
although, i had a good time there.   Ive heard that theres some Islands
off of Georgia but dont know the name of them. 

Can u recommend other Islands of interest ?  I must take my vacation in
March , APril , or early May .... and would prefer outdoor temps. at
least in the 50's.
Although im not rich, the cost is secondary .

Ill check back here every couple of days for replies.

Thanks much,
Dave from Illinois.