Are you on any medications? Tons of medications cause dry mouth and throat, leading you to drink more water, leading you to urinate more. Some medications, such as allergy medications can cause headaches and even back pain. I had to stop taking Allegra because it was giving me lower back pain. Read the information about the side-effects of any medications you take, even over-the-counter stuff (look online for the side-effects of those, not just on the bottle).
I agree that you should also make sure you are getting enough electrolytes with your water. Sodium, potassium, calcium, etc. If you urinate an hour after you drink water, this doesn't mean you are urinating out the same water. Your body is trying to keep things balanced. Is the urine usually really clear or light yellow, or is it dark yellow?
If it's defiantly not any medication you take, I think it would be prudent to have your kidneys checked. I agree that a urologist would be a good next step. Have they done a urine test?
Calcium deficiency doesn't really cause the symptoms you are describing. It causes muscle spasms, cramping, twitching, and slight numbness (usually starting in the lips). Plus, both blood tests (although different) showed results in the normal-range. If you had something that caused a calcium imbalance, you wouldn't be normal just for the tests.
I hope you can get a diagnosis soon. Good luck. Let us know if you have any more questions. Also let us know what the doctor says. It may help someone else on here in the future.