Catrin, keep practicing whatever movements you think you need to have before you do any major adjustments, changing of shifters, etc. I'd hate to see you spend a lot of $, when this is more of a practice issue.
I am not sure it's easy for some to understand how it feels to not be able to take your hands off of the bars, or change position. I've been riding for almost ten years and I still have a very difficult time with the water bottle and will only drink while riding if the conditions are what i term "optimal."
I had a lot of issues using the drops until two years ago. There were no issues with the STI shifters, etc., that always seemed intuitive to me, even when i switched from trigger shifters, when I got my first road bike. But, I always felt like I was almost lying down when trying to ride in the drops. In observing my DH, I noticed he looked like he just moved his hands down, but his body stayed in the same position when switching to the drops. So, I knew the shape/position of my bars had something to do with it, but most of it was my shaky coordination. I knew my core was in pretty good shape (maybe not as good now ), so I did 2 things. First, I bought the short and shallow bars. Did tons of research and narrowed it down to 2-3. I bought the Specialized ones, but there are others that would have been fine.
Then, I just made myself practice. The bars made a difference, but I found that by getting in the drops on flat roads, where I could put it in the big ring and go really helped.
I still don't feel as confident with my handling skills when in the drops, but I use them a lot now.