If persons don't know each other too well or have used the "Sie" for some time, it's an unwritten law that typically the older person offers the "du" to the younger one by saying something like
Ich bin "Vorname".
The younger answers similarly, followed by a handshake1) or a toast2).
The sentence quoted above
Darf ich Du sagen?
is typically used by older persons interacting with teenagers (teachers, doctors, etc.), where it's unclear whether they are of age already or whether the younger cares. It typically does not imply that the asked person is to say "Du" to the asker. The "Sie" is generally on the retreat, and it's very common that people of the same age or within a group or in a company just use "Du".
Darf ich Sie dutzen?
is correct, it's rarely openly asked. Either you offer the "Du" (by introducing yourself with the first name) or you somehow have the feeling that it's ok to do so.
1) unless COVID forbids so
2) if in a restaurant or a pub