Many folks find great success with switching to left hand shooting when left eye dominant even though right handed.
Red dots can be great with bows. It lets you focus on the target and have the dot be in focus at the same time. You can even add some magnification on some sights and change the reticle shape/type and brightness to suit your needs. If you shoot the same distance all the time then you can use just about any red dot sight out there that you can mount to your bow.
The trick comes when you try to have an adjustable red dot sight for varying distances and how specific you want to be dialed in for a particular distance.
There is one manufacturer that I know of right now - ARD - Adjustable Red Dot that has solved some of the problems with needing to use red dot sights on bows at varying distances. Their successive models and prototypes are evolving to help with getting finer gradations for specific distances, but the engineering tolerances that are required make the effort expensive and difficult to machine. Their current products are great for hunting. I have tried ARD for use with 3D, target, and field archery and just couldn't get the desired effect where I needed to dial into a specific yardage. If you are a gap shooter and you get the sight dialed for several known ranges like 18, 26, 32, 41, etc whatever your setup will allow and you are comfortable with that, then it works great.
Shooting a red dot on a bow can also make it easier or more forgiving in some circumstances to hit what you're intending without having perfect shooting form or bow setup.