PROBLEM: Your bike veers and feels unsteady when you take a hand from the bar to reach down for your bottle or back to your pockets.
SOLUTION: Before reaching, move your other hand to the bar top next to the stem.
This puts your grip on the bike's centerline where steering leverage is minimized. Then, inadvertent movement caused by leaning or twisting your upper body won't make the bike waver very much.
Experiment in a safe place and you'll see.
First, grip the drops or brake lever hoods. Then reach down for your bottle. Notice some instability?
Next, move one hand next to the stem, bend that arm's elbow, and reach down. Ah, much better!
Bonus! When you're using this grip, your front wheel is less likely to be knocked off course if it should hit a rock, bump or road flaw while you're riding one-handed.