Without knowing what MTA you are using to send mail from your droplet (postfix, sendmail, something else) it's hard to give you an exact fix for this. This page has a configuration option for sendmail which should resolve the problem.

Generally the problem will happen if your web server's MTA beleives that it is handling mail for your domain rather than the outside service. Because of this, when it receives an email addressed to someone@yourdomain.com it will try to deliver it locally instead of relaying it to the service defined by your MX records.