There are a couple on the San Sebastian River right there in St Augustine.
Oasis Boat Yard And Marina.
St Augustine Marine Center
Xynides Boat Yard - gets good DIY reviews.
But also have a look at this facility, right on the St John's with easy access.
St. Johns Boat Co. - Home Page
As for a mooring. There is now a good solid mooring field in St Augustine. Right on the ICW with great access to the Atlantic. Mooring both sides of the Bridge of Lions. Managed by the St Augustine City Marina.