Me too me too me too me too! The exact same issue as MarcMenninger's, & I also noticed it yesterday (3/22/19), so whatever change could be causing it, it must have happened in the last couple of days. It is really annoying, as it has been working for years
without a glitch, and now all of a sudden, without any warning, it just stopped working.
To give even more details; as OP described, the sent mails are just sitting in the Drafts folder with the very obscure "Something went wrong..." message near the top. Then, if I open one of these mails for editing & try to send it again, a small error window
pops up, and when clicking on "More information", it gives a little more helpful error message: "You don't have permission to perform this action." Now we are talking! So it seems that whatever the outgoing server is (probably not the original send-only SMTP
server, but an Outlook server), it refuses to recognize the send-only account's email address (an email alias for Outlook.com) as a legit sender.
Microsoft team, please try to fix this issue quickly. I've been quite happy with Outlook.com, & would hate to say good-bye to it. Thanks!