Low-key US weekends under $5000?

My fiance and I are looking for a low-key, relaxing honeymoon.

We were thinking Hawaii or Puerto Rico, but Hawaii is expensive and we can't find much in Puerto Rico that isn't very group-oriented.

We'd love a combination of indoors/outdoors, a nice meal, some fun activities.

Where can a US couple go for a relaxing, but unique, honeymoon weekend as people without a lot of money to burn?