First off, do NOT shoot between the 2 targets. As you've already seen this produces horrible results. What you need to do is train your MIND to automatically select the correct target. In your case, the target on the LEFT is produced by your Right Eye. So, if you want to continue to shoot using your right eye then shoot at the left target. BTW, you can confirm what I'm telling you will work by "winking" your left eye when your lined up on a target.
Note, I'm another with what could be called "mixed dominance". In my case it a matter of using my left eye with SLR cameras for 40 years and shooting handguns right eye dominant. In addition to mixed dominance I have also lost the ability to focus on handgun sights, so my shooting today is target focused. As a result I see one target and 2 handguns. For me the right handgun is correct for shooting with my right eye.
Now for a bit of a kicker. I can aim and fire 2 handguns at the same target point at the same time because of my mixed dominance. I can also switch to my left eye when shooting left handed and because my vision is a bit better in my left eye in slow fire I'm usually a bit more accurate "weak" hand.
To sum it up, when you find your turning into an Old Dog you'll find it's helpful to teach yourself a few new tricks to compensate for the changes in vision and things like that.