Hi ~ Welcome to the board.
One thing comes to mind since a UTI is not the cause (and, no, dripping urine is NOT normal):
If your dog is eating too high a protein food, that can cause incontinence....especially nocturnal incontinence. The food doesn't have to be a high protein food...just too high for him. (I had this happen with both of my Akitas...male and female...and all I had to do was change their food to one with a lower protein level.) The test for protein in urine is a specific one....not one that's done with the usual urinalysis.
If your dog all of a sudden was drinking more water, I'd say that was probably the cause. But if he's not drinking more than usual, I'm less prone to think it's that.
If he has dripped urine all of his life, he may have a structural problem or something going on with his prostate.
Didn't the vet want to pursue anything further?