Personally, I think that we should be carrying 5 running backs, each with designated roles. In that group of 5, there is room for a fullback who also plays special teams.
We need two starting quality running backs (thunder and lightning), and a 3rd down specialist. That leaves 2 positions. The roles for those positions include backing up at running back, backing up at 3rd down back, short yardage specialist, and blocker.
A true fullback would take more snaps than lots of others on our team. The number is somewhat like TE numbers. The number depends on who we have in the position. I don't have the reps for other teams, but someone might check the jets.
All this being said, I don't think the draft is where to find fullbacks. Fullbacks out of college have too much to learn about blocking and receiving. They are used to carrying the ball.
BOTTOM LINE
Belichick likes our pasing oriented team, with no emphasis on the running game. The running game does happen and we do well only when the run is set up by the pass. A fullback allows us more of a commitment to the run when game plan or game circumstances dictates that this is what we need to do.
We are passing team and should remain a passing team. That being said, we need to improve our running game.