Couple quick thougths. I returned to biking last year and have logged just under 200 miles. I used to ride a bike daily for 5 years. I expected balance and shifting and handling to be no brainers. Not the case. The muscles don't work the way they used to. The bike handles totally different. The shifting is different. The rider positioning is different. The tires are different. The muscles reach tired situation quicker and so tend to have jerkier movements.

It sounds like you are making good progress in returning to biking. My guess is that you just need more time in the saddle, so all this stuff becomes literally nobrain situations and the muscles have an opportunity to rebuild.

I've noticed the most difficulty in the middle position,i.e. on the levers. If I'm in the drops, I don't have too many stability problems. If I'm up on the bars, I also don't have problems. But the levers still feel a bit funny and the wheels wobble more than I would like. This month is better than last month, which is better than previous. So I think it is more of a conditioning question.

Your setup sounds good. If you want to try a change, you could ask your LBS to swap the 700x23 for something like
a 700x28 or 700x32 tire and see if that makes any difference. Another option would be to see if you could mount the water bottle higher on the frame so you don't have to reach so far.

Good luck and good pedalling.