I also have spent quite a lot of time trying to train my off-hand to be more dextrous, in not just cubing too. I started in middle school where I would write my ABC's with my left hand (I am right handed) over and over until my hand hurt. Now I can write somewhat comfortably with my left hand, only my handwriting is atrocious (not that it's much better with my right hand).
For cubing I started in 2003 after the World Competition in deciding to train my left hand. I simply reflected my algs over to lefty and did them over and over and over and over and over. At first it felt incredibly awkward to do the algs left handed, but over time it felt easier and easier. Now my left hand is faster at fingery style trigger moves than my right hand, which does more of the hard triggers like (R U' R' U). My left hand is actually faster now on average at doing the T perm and the J perm. I feel like I can do sunes with right or left at comparable speeds. Most, if not all, of my 2 gen OLLs and COLLs are faster with my left hand, but I do them righty or lefty now. I'm also learning to do the A perm lefty, which I've never really done before, this way I don't have to rotate during PLL for that case. I currently do all edge PLLs (except H) righty or lefty and I'm almost to the point where I can do the MU alg Z perm righty or lefty at comparable speeds.
You just have to simply decided that you will stop having a dominant hand in cubing. Then just practice practice practice. It's as easy as that.
Hope this helps,
Chris