I have had some issues with this as I'm a stroke survivor. I had paralysis on the left side and my balance has never been 100% normal since then, although it's improved tons.

It took me quite a while (months) to be able to grab/replace bottles while I was riding without weaving all over the road, and even longer to be able to ride no hands. A couple of thoughts...

This might sound counterintuitive, but are you looking down at the bottles when you're trying to grab them? IME, this will make me miss/swerve. I have to keep looking down the road. Wherever you are looking, is where your body and then the bike will tend to go. Also, shift your weight back some. You don't need to see the bottle or cage to pick it up or put it back, you just think you do.

If you really do have bad balance, and it isn't a function of your bike, yoga helps a lot. You might feel stupid doing it, but that helped me more than anything else when I was recovering from the stroke. You might consider it.