"Hi. We appear to be close DNA matches. I am trying to understand my biological family history. Would you be open to comparing family-tree information?"
"I recently learned information about my adoption and am trying to confirm family connections. I understand this may be unexpected."
"I am not asking for anything public. I would be grateful for medical-history information if you are willing to share it."
"If you are not the right person to talk with, is there someone in the family who handles genealogy or family history?"