I'm at the point where I don't even notice I'm doing it anymore, and I can run a rack offhanded. This is the secret that got me there:
Just force yourself to do it when the situation calls for it. Do it instead of using the bridge as much as possible.
If you do this, you won't even need a special opposite handed practice session... you'll find that it comes up in your daily playing all the time. You'll do it like 5-10 times a night every single night you shoot pool. I know it can be tough to find the will. Many times I've suggested it to 'trainees' and they balk and get out the bridge anyway.
....But don't be like that, MAKE yourself do it. Don't worry about missing the ball, even an 'easy' one. Be prepared to miss a bunch. Eventually you'll get better at it and realize why it's so much better than using a bridge.
As for bridging with your dumb hand... I just use a standard closed bridge always (even tho I'm mostly open with my usual hand). I think it helps keep the awkwardness of your stroke in check and prevents the tip from wandering a lot. Work hard at not steering or muscling the ball, strive for a simple back and forth motion when you're lining it up.
One more thing that's worth mentioning: The offhanded shot is a very specific shot where the cue ball falls in a certain limited area and the object ball does too. You're not going to need to shoot offhanded into the side pocket, for example.
For this reason, I recommend you don't waste time learning to run racks offhanded. It's fun to show off for the girls but you're training yourself to make shots you'll never ever need to make in a real game. Better to devote that training to the common situation where you WILL need to use it - when you fell a bit too thin on a shot that runs parallel the short rail, and the CB is like 1 to 1.5 diamonds away from it and near the centerline. If you want to do a custom practice session for this, set up shots you really can't reach and would use the opposite hand for anyway. Don't bother with shots you can reach or shots that only work with a bridge.