Oleg Volk discusses some strategies for overcoming cross-eye dominance over at http://cheaperthandirt.com/blog/?p=14573.
I am not left-handed, but I am left eye dominant. I was never able (and still cannot) close my left eye while keeping my right eye open. When I started shooting as a boy, I simply switched hands and learned to shoot left-handed. Decades later, and I still shoot like that. It’s worked out just fine for me.