I see that I can include a second account of mine in each conversation but was wondering if there was another way. Thanks
I don't think there is another way, unless you want to use their API and keep track of all tweets from your account. If you use their API, you can even store them all in your own database.
You can probably write a bot that will send/store the messages https://developer.twitter.com/en/docs/direct-messages/sending-and-receiving/api-reference/get-event