You don't need to have a domain associated with a droplet in order to send emails to external mail services. I suggest you take a look at the logs of your MTA (Mail Transfer Agent) to see what's happening when the droplet tries to send an email.
You haven't mentioned which exact MTA you're using on your droplet (exim4, Postfix, sendmail, etc.) so I can't point you to the exact log location, but you'll likely find what you're looking for in
/var/log where all logs are usually stored. The logs will give you at least a general idea of what's the root cause of the issue and will make further troubleshooting easier.
Also, it's important to note that the emails you send from your droplet after you resolve the issue will likely start ending up in the Spam folder on the receiving end since you don't have proper SPF, DKIM & DMARC DNS records for your droplet. There's really no way around that other than manually marking the emails as not spam in an attempt to train the spam filters to start letting the emails coming from your droplet's IP go to the Inbox folder.