I can't offer any suggestions on pool rooms in Owensboro but I will tell you if you like BBQ, you have to go to Old Hickory. IMHO, one of the best in the country. I'd recommend the 3 meat sampler and save room for the blackberry cobbler ala mode. It is to die for. Some people may want to send you to Moonlight, which is good, but no where near as good as Old Hickory.
If Owensboro doesn't have anything to offer in the way of pool, Evansville, IN is about 1/2 hour away. A little larger city and should have something in the way of pool.