"Your draft message was not copied to your drafts folder (Drafts) due to network or file access errors."

This isn't a certificate error. Why do you think your email provider has an invalid certificate?

Annoyingly, it also seems to prevent me saving the message I just wrote into the Drafts folder.

As a workaround you can select the Drafts folder underneath the Local Folders account as Drafts folder for your IMAP account. Check your Account Settings - Copies & Folders.

Two separate issues.

An invalid certificate was preventing Thunderbird from connecting to SMTP & IMAP.

Separately, I cannot save to the mailbox Drafts folder any time I can't connect to the server. Why not? Shouldn't the message get saved to the local replica of the Drafts folder, & sync to the server next time there is a connection? Just the same as other synced folders?

An invalid certificate was preventing Thunderbird from connecting to SMTP & IMAP.

Was? So the problem is gone now? If not, please post a screenshot of the error.

Shouldn't the message get saved to the local replica of the Drafts folder, & sync to the server next time there is a connection?

It's the other way round. The message is uploaded to the server, and then synced with the local email client.