As riders it is important that we have great coordination on both sides of our body. I am right handed so I was always right hand dominated when I was riding. My brain would tell my left hand to do something but my right hand would take over and try to do the job. This has a down side to it.
How I started to overcome this was creating better coordination with my left hand. I started out by just doing simple tasks like when I would go to open a drawer or a door I would consciously use my left hand. Then as I started to gain more coordination I would do some harder tasks, like brushing my teeth with my left hand. Just by getting my left hand used to doing more things it became a lot easier under saddle to get my left hand to do its job.
So start practising everyday tasks with your less dominant hand and watch how your riding improves!