This same topic has been gone over again and again here. It always comes down to the same thing: The reason the Navy asks for LORs is not to find out how well connected you are (by the way, an army O-3 is not going to cause anyone on the Navy selection board to wet themselves). The point to the LORs is to get an idea of what kind of person you are. Mine were from one of my old high school teachers, a Motorola engineer (who was actually unemployed at the time he wrote my LOR), and the GM at the restaurant I worked at. No one who's name you would drop at a cocktail party. Obviously it worked out OK for me.
Just get letters from people you know who can actually attest to your character. Work on the quality of the letter, not the status of the writer. Also, I agree with Steve. Don't assume that they are going to do it well. I had all my references show me a copy beforehand so that we could be sure it was what I needed. Not a bad idea, but try not to write (that's "right" to you, Tom) the letter yourself and have the person sign it. They'll see through that. Good luck.