Question #1:  Does the existing home need to be sold?

ANSWER:  the new home must be their principal residence, so while they do not have to sell the old home, you can only have one principal residence, so unless they rent out the old home, or convert the old home to a second home it will need to be sold

Question #2:  For a married couple, do both need to be on the deed?

ANSWER:  They do not both need to be on deed – again it just needs to be their principal residence and they need to file a joint return; (lender requirements may trump this).