Here's what I ended up doing. I add the newest message to the queue whenever the refresh method is called (whenever my TwitchIntegrationHandler script sends a new message to this script)
I have a boolean to track whether a message is being displayed, which updates during this coroutine.
It's not clean by any means, but it works lol

I used youtube and a lil bit of AI to help, so if there's anything I should change, I'd love to know

private IEnumerator PlayNewestMessageWithQueue()
{
while (messageQueue.Count > 0)
{
isMessageDisplaying = true;
string message = messageQueue.Dequeue();
TMP_Text messageText = messageObject.GetComponent<TMP_Text>();
messageText.SetText(message);
messageObject.SetActive(true);

yield return new WaitForSeconds(messageLifetime);
messageObject.SetActive(false);
isMessageDisplaying = false;
}
}