Is it possible to get a room for about $125.00 to $150.00 per night?
Of course, it's Sicily. Granted Siracusa is very touristy, but you should be able to if you book early enough (which means now because B & Bs are small and they fill up fast).
B & B Porta Marina (http://www.portamarinaortigia.it/) - you can get a sea view on an upper floor. Only thing to keep in mind is is a mom and pop property with stairs (no elevator that I recall) but in a great location.
There are tons of other options on www.booking.com, just have a look and narrow choices by your price range.