Im sure there is a text book way to shoot opposite unsupported hand with the cross dominant eye. The competitive pistol shooters could help you with this. But Ill give you the method I used when I shot action pistol regularly.
First of all the main thing most people lack is coordination of motor skills with the weak side. In your daily activates focus on using your weak side more, small tasks such as: reaching for things, taking money out of your wallet with your wallet, brushing your teeth etc all with your weak side. After a while you will develop pretty decent coordination.
For strength exercises buy one of those hand grip exercisers or a racket ball and squeeze this while you are driving, walking around, watching tv
Other than that go to your kitchen and grab a heavy container of some sort of canned food. Use this to do wrist rolls and also use this buy standing up with your arm straight down by your side with the can in hand and raise your arm in front of you to line of sight.
When actually shooting using the weak unsupported side keep your arm straight, wrist locked and grip your pistol high and firm enough that the skin under your fingernails turns white. Shoot like this for a while and over a period of time you will be able to adjust as you gain a little confidence and coordination. When lining up the sights bring the gun in front just like the can of soup, and to the center of your chest then turn your head very slightly to the weak side so that your dominant eye is focusing on the front sight and the target.
This may not be the text book method but it worked for me.
Be Safe,
Scott