Selling and buying -- how to time it right?

First, make the sale of your house contigent on you finding suitable housing. Then when you do have an accepted contract and your new house under contract, get addendums signed that state the new owner cannot take possession until you have had the opportunity to move your animals to the new home. Give a set amount of days after closing, the shorter the better. You may have to compensate the new owner for a few days rent, but I'm sure you can work something out.