Practice just having one hand off the bars doing anything: reaching in a jersey pocket, resting on the top of the bottle in the cage, cleaning your front tire while rolling (GLOVED hand necessary). You need to be relaxed yet have some core control. This may also highlight any issues you have with your bike fit--you shouldn't need your hands for support, but it doesn't mean you need to be a master at riding no-handed either.

The thing that help me get the hang of drinking and fiddling with bottles the most was to start out practicing on flat ground. Downhill, and you may need the bars. Uphill, and the effort may cause more swaying. It also means that you can coast for a second if you need room to get around your leg with the bottle, and you won't slow so much that you'll get wobbly. Shift a cog or two harder than how you normally cruise, especially if you hold a fast cadence. This allows you to get some stability from your legs and the resistance on the pedals. Second, try to retrieve and replace the bottle while pedaling. Keep everything normal and just glance down if you have a problem. Because you've already tried taking your hand off the bars and doing stuff with it (including messing with sunglasses, helmet, or anything that requires your hand to be up), you shouldn't have a problem taking a drink. If you do, try to grab the bottle as if it's a baby bottle so that you can get the bottle turned enough to drink without leaning your head around. Keep pedaling and looking ahead to replace. Shift back to cruising gears. Practice, practice.