I'm not entirely sure about this, but isn't the edit ability of Telegraph based on what I believe is a cookie, it seems to be called "tph_uuid" and you can see it if you press F12 in Firefox, go to "Network", reload the page, find the single POST and you can see it in the request headers.

Meaning, as long as you don't clear your cache you should be fine, and if you do, maybe you can back it up somewhere.

Sounds like what that site really needs is the same thing that almost every other major site has -- accounts.

You can use @telegraph on Telegram to save/edit on your account.