Can I sign my mortgage over to a family member

No you can't sign your mortgage over to a family member.

If the existing mortgage is an FHA loan, there is the possibility that they could "assume" the mortgage... however they would have to "qualify" for the loan on their own.

This depends on the mortgage. YOU can't just sign it over. However, the lender can allow the family member to assume the mortgage. Such a clause would have to be written into the note or be agreeable to the lender. They would have to qualify as noted.