I would not try drinking. Okay, it might help you sleep, but it is not a long-term solution. You may drink more once you become more tolerant, and that will end up doing you more harm than good. It could lead to other health problems.
Is the T keeping you up at night or is it the anxiety? Have you tried playing sounds to distract your mind from the noise. That worked for me when I had difficulty. Or perhaps try medication, but only after speaking with your doctor.
Usually if we are not anxious about our T we don't mind it. I remember that even the sounds I used to mask kept me up. So one night I forced myself to think of other things. My mind would go back to worrying about the T, and I had to actively think, "No, don't focus on that, think about this..." It was a hard task, but it worked and I managed to fall asleep. I now fall asleep without maskers. I think my T is probably lower than other people's, but I have stopped worrying about it (most of the time anyway...sometimesit does get me down). Once we stop getting anxious about it, we tend not to be bothered about it when we're sleeping.