After hearing the history I started out with a physical exam and pretty soon felt like I had found the possible culprit. Male dogs have a skin fold called the prepuce (similar to foreskin in humans) that covers and protects the penis. In this case, the opening of the prepuce was very narrow and there was a large amount of extra skin. I thought that the narrow opening was leading to the urine pooling in the extra skin and then dribbling out after the dog was done urinating. I suggested that we perform surgery to remove the extra skin and make the opening wider.
The owners agreed and we performed the surgery the next week. I called for an update a couple days later and the owners reported that the problem was resolved and no more dribbling was happening. If you would have told me in vet school that I would essentially circumcise a dog I would have laughed, but like I said earlier, I am always surprised at what I get to see and do as a veterinarian.