I asked my coach the other day about this and I was told that if you start a tournament or even weekly leagues bowling 2 handed and then changed to 1 handed it is legal as long as you bowl from your right if it is a right handed average. The same if you had a left hand average.
So there for the ball need to be delivered from your right side and that is still legal, weather you bowl with 2 hands or 1 hand. Provided it is a right hand average. Or if you had a left hand average you would need to deliver the ball from you left side.
I also believe that if you start right handed you need to finish right handed, and so forth if you start with your left hand.
Just what I was told as it came up at league one week and I wanted to know the correct rule with this.