No way! My wife says I’m useless at multi-tasking (although I’m not really as bad as she claims).
Fortunately, CW QSOs are simplex so I don’t need to. In fact, I wonder if anyone could do it because sending (translation of sentences into Morse) and receiving (translation of Morse to sentences) surely involves the same part of the brain (like trying to talk and listen simultaneously).
The two tasks don’t overlap so it’s no different from what you do. Of course, I wouldn’t propose an experienced operator (like yourself) re-train to paddle with the other hand for a small efficiency benefit.
But, for a new operator looking to use a paddle for the first time, it’s worth considering teaching yourself to use your non-writing hand (which is otherwise not doing anything useful – unless it’s holding a glass of beer). After all, iambic paddling only involve your thumb and forefinger and, as I said, I had no trouble learning to use the opposite hand.