Hello tiburtine,
It seems like you're experiencing an issue with email drafts disappearing on your MacBook Air, and they appear on another one of your devices, correct? We're here to help out.
It seems like you're using the Handoff feature. "If you use Handoff, you can start your message in Mail on another device, then finish it in Mail on your Mac. To continue writing a message handed off to your Mac, click the Handoff Mail icon that appears at the left side of the Dock."
You can find more about writing emails here: Write and send emails in Mail on Mac
For more details about the Handoff feature and how handing off between devices works, please check out this page: Pick up where you left off with Handoff on Mac
The above link also outlines how to manage the Handoff feature or turn it off if you choose to.
Take care!