Body not absorbing water goes to bladder

I have a strange problem. Whenever I drink water, I feel the urge to urinate within half an hour or so. I feel like water going straight to bladder and not sure if body is absorbing any water. Moreover bladder doesn't become empty after urination. I need to go 2 or 3 times to empty fully. Then I feel thirsty again and drink water again. This bothers me more in the night before going to bed and also when the weather is a bit cold. Will this frequent urination damage my kidneys? I have been facing this issue since long. Consulted urologist, few tests done. Found mild prostate enlargement (BPH). but other than that no other red flags. Used medicines but not much helpful. What could be the possible reason for this problem? Do I need to go for any further tests to find out what is causing it?