Your address never changes
Your DatsMe address is minted once when you join and never changes afterwards, so any link to your page keeps working for as long as the page exists.
When you sign up, DatsMe builds your address from the name you give it: lowercased, with anything that is not a letter or a number removed, followed by a number that makes it unique. Two people called Sam become sam.1 and sam.2.
You do not choose it, and you cannot change it later. That sounds like a limitation and is closer to a guarantee: every link anyone has ever saved to your page, every message addressed to you, and every reference to something you published keeps resolving. Nothing quietly breaks because you renamed yourself.
Your display name is separate and you can change it whenever you like. The address is the identity; the name is what people read.
The same address is your mail address with @datsme.me after it — see DatsMail.