Oh...another thought I forgot to mention. If his urine is "dribbling" then it is just possible he isn't emptying his bladder properly. Over time, this can cause problems, usually bladder infection, which, if left unattended can possibly re-locate to the ureters, and kidneys. You could have a word with your vet about this, and the vet might (?)recommend a course of antibiotics just to be on the safe side.
OK so he's drinking normally, no increase, so that is a very good sign as far as his kidneys go.
This sounds very much to me like enlargement of the prostate gland. Get the vet to check his prostate. It's possible he will be able to feel the shape of it and whether it is enlarged from a quick anal exam
If your dog is fit and well, and doesn't have any health issues which might make surgery risky -then the best option, even though he is 12, might be to go for having him neutered. 12 is just "getting on a bit" as far as average Jack Russells go. They can live to a ripe old age.