Find a deck of playing cards and shuffle them. Swap the deck to your other hand and shuffle them again. Unless you are ambidextrous, this may pose some difficulty. Persist. To help, swap back to your natural way of holding the cards and watch how your fingers move, apply and release pressure whilst shuffling the cards. Swap hands and try shuffling again, it should get easier the more you practice.
In rehab, I constantly have to undertake activities with my left (non-dominant hand) as it’s affected by the stroke and my post brain surgery weakness. The goal after stroke is to regain strength followed by controlled movement. Having regained strength in my hand, I now need practice activities without focussing so much on my hand’s functioning (as my brain insists on telling my fingers to over-grip, making many activities impossibly). By focussing on activities rather than hand function, I effectively trick my brain into not sending the over-grip signal because it is distracted; hence the second part of the challenge.
Next, deal the cards out and play a couple of rounds of Solitaire or any other single player game, using your non-dominate hand to deal and turn over cards.
If you want to continue the experiment of using non-dominant limbs, you will find your grooming routine holds numerous opportunities. I would however caution you from starting with brushing your teeth with your non-dominant hand as toothpaste in the eye, stings, a lot.