What youre experiencing is a very common complaint. As we age, several things happen. One is some enlargement of the prostate gland. As the gland increases in size, the bladder (which stores urine) has to exert more pressure to get rid of the last drops of urine. When that happens, some of that pressure gets placed on the part of the bladder called the bladder neck.
This muscular portion of the bladder can contract, leading to the stinging sensation you mention. Many men will state that the stinging sensation seems to be located at the very tip of the penis. If so, bladder neck pressure is the most likely cause.
However, you can also feel stinging sensations with either an infection in the bladder (urinary tract infection) or in the prostate gland (prostatitis). To be diligent, you should see your doctor to have your urine evaluated.
You raise the question of an STD. Its most unlikely. But its good to discuss this with your doctor. He or she can check for an STD at the same time you submit a urine sample to look for a urinary tract infection. However, a 32-year monogamous relationship (and your partner attests to this as well) and the absence of any milky or whitish penile discharge would argue against an STD.
The dripping sensation can also be a symptom of an enlarged prostate gland or an overactive bladder. Less likely, but important also to consider are prostate or bladder cancer.
It would be wise to ask your to doctor perform a digital rectal exam. This assesses the prostate gland size and texture. If your prostate is enlarged, your doctor may suggest therapy for this condition.
Quite often, doctors dont find a specific reason for the types of symptoms you have. And initial treatment might not help. Certainly, this can be frustrating. But keep working with your doctor. Its likely you will get some relief.