You've gotta run the numbers to make any kind of fair comparison. See what you could reasonably charge for renting your house out, cut that in half (see: "50% rule), subtract your mortgage payments, that is your net income (or loss) from renting your house to others. Subtract your net rental income (or add in the case of a net loss) from the cost for you to rent yourself, and compare that to your current housing payment.
Make sure you consider all your housing costs, such as maintenance, utilities, etc.
Without knowing any of the numbers, I'd venture to guess that continuing to hold your house will end up being a better choice, especially with pets. Plus, you can be assured your house payment will not go up, whereas rent certainly will.
That's the financial side of the decision. Only you can weigh freedom vs. certainty, whether you're prepared to be a landlord (it's no picnic or free lunch), whether you can handle a tenant destroying your house and fleeing, whether you're prepared to move again if your own rent is raised 20% all at once, etc.