Most services will serve custom domains. Using DigitalOcean you'd of course have to run your own mail server. For the domain part you'd likely just have to upda

Most services will serve custom domains. Using DigitalOcean you'd of course have to run your own mail server. For the domain part you'd likely just have to update your domain's name servers to DigitalOcean's and then add at least one MX record pointing to your current provider until you switch. When you switch just update the MX record to point to your new mail server.