In order to use a custom emoji as part of a response node (such as in an embed description or message content) you need to use the format: <:name:id>. It is actually very easy to find this information for an emoji.
To find this string for a certain emoji first go to the Discord server it is in and do the following:
Type \:EmojiName: and click enter to send the message. In my case the emoji I'm finding is :WhiteDBSIcon1:. So I would type \:WhiteDBSIcon1: and press enter.
After you do this, Discord will automatically convert your message to the <:name:id> format we need:
Then you can simply paste this string anywhere in a DBS response, like a Send Message node's content, or a Send Embed's description field:
This will then appear as the actual emoji when the response is sent in server.