I have the same problem but I know exactly what I need - INCREASED CORE STRENGTH.

I was having a fitting done a few weeks ago and I mentioned that it felt like there was a lot of weight on my hands. While pedaling, the fitter asked me to let go of the bars and keep pedaling. I fell straight forward - not enough core strength to hold myself up. He said I should be able to let go and keep pedaling without moving my upper body. So I got a Swiss ball the other day and am going to start working with it.

My goal is to be a better, stronger cyclist, and I guess this is just the next step in the process.