My wife used to have this problem. Except, instead of stopping to drink, she just wouldn't drink. Which is much worse.

Anyway, I didn't see anyone mention this, but one thing to do to make it easier is to stop pedaling and coast for a minute while yoiu grab your bottle.

As for learning to ride no handed, shift in to a slightly harder gear so you aren't spinning too fast, and slide forward on the saddle, so the front of the saddle is lodged a little between your ass cheeks. It will give you a little more balance.