<< Plus, when I floss, there is a pretty intense smell between the teeth. >>
There are two reasons for this: you don't floss regularly enough to get food particles out before they begin to decompose, you have a few cavities (tooth decay), or both.
I had a similar problem, I have pretty good oral hygiene (brush, floss, gargle) and all so I didn't think it was a cavity, plus the tooth wasn't sensitive or anything. I went to the dentist and he found the tooth had started decaying below the gum line and it was on the verge of abscess. He also found two other cavities located between my teeth.
I had the bad tooth pulled, he wanted to crown it (root canal) but I didn't have the $700. The other two were drilled and filled plus the I got a good cleaning which I hadn't had in two years (complete with that gum line scraping everyone loves).
I tell ya, my breath is 200% better. I didn't realize my breath was going sour because it happened very gradually. It wasn't like I woke up one morning and WOW!
Moral of the story...go to the dentist soon.