I had the same problem strumming with my left hand. I had always finger picked but finally conquered the beast.I've always used both hands to shave, brush my teeth, lock and unlock doors etc. I'd say I'm probably 55 to 60% right hand dominant though.
Shaving with either hand is possible but it takes practice to build muscle memory. I'd also recommend using your non dominant hand for minor tasks such as drinking your coffee, opening doors to get your brain used to controlling your off hand.
@RayClem, I cannot strum guitar with my left hand, but can make chord shapes with my right. I can play bass left or right though. Strumming is just too awkward.