I currently use GoDaddy to register my domain names, but Digital Ocean handles my servers and DNS.
From your DigitalOcean control panel, you can add the domain names that you want and set-up any DNS records that you need. You’ll need to set-up one for the wordpress site to point to your digital ocean server. If you have any email address set-up with your account, you will need to be sure to set-up that DNS information as well.
Then I went into GoDaddy’s Domain Manager and got rid of all the default records, and then added the following name servers:
Wait 10-120 minutes and the domain name should be switched over. (it says it can take several days, but I have never seen it take that long.)