singing F against D7 chord

I heard a recording of my band and realized how much I was singing flat, especially on notes in the middle of my range. I'm trying to go over each song slowly and really nail down the intonation. It doesn't help that I'm drumming pretty hard and trying to hit some accents while singing.

Anyway, one newer song I am doing is "Down in Mississippi (and Up to No Good)" and at the beginning of each verse it sounds like the singer in Sugarland is singing F while the band plays D7. This is just a bluesy flatted 3rd, right? I'm not very knowledgeable about harmony.

I think I "got" that okay but with the band something seems wrong. If they don't play a good strong D7 it feels strange. Should they be playing Dm7 or something else to help me with this? Or is it some more exotic voicing like D7/9 ??

Sounds odd to me. In the blues scale the flatted (or minor) third replaces the major third; the blue note is the flatted fifth, which is added to the fifth. So in the blues scale playing a flatted fifth above a major chord sounds, well, bluesy; but playing a minor third above a major third--that will just sound dissonant.

Or so it seems. I'm not near a keyboard right now so I can't try it out.

F against D7 would sound dissonent IMHO...

normally singing an "F" against a D7 chord would mean you are singing a minor third against a Major third in the chord. This can also be interpreted another way. Singing an "F" against a "D7" Chord is also singing a (#9). This is the chord that you hear in "Purple Haxe" or "Foxy Lady"