We went last July for just a few days. Had a great time at the Pirate Museum. We were on a budget too. I rented a home through AirBnB.com and paid $95 a night for the whole house. It was a block from the beach. We cooked meals in when we could.
As for when to go, I would suggest as close to October as you can get. Summer down here is brutal. And St Augustine is the kind of place where you park your car and do a lot of walking. So I would keep that in mind.