It should be possible to configure your DNS settings to continue using the GoDaddy email service. All you need to do is duplicate the records for mail related subdomains that exist (mail,smtp,pop3,imap) and set up the same MX records as you had on GoDaddy.

Unless GoDaddy policy prohibits doing this, I would recommend reaching out to their support team as they should be able to provide you with the DNS settings you’ll need to add to your domain on DigitalOcean.