@Ashley Mills - It sounds like your new email address may not be fully connected to HoneyBook's email integration, even though you've updated the email associated with your HoneyBook account. The fact that emails sent from your old address are still appearing in the project activity but emails from the new address aren't is a pretty good clue that the old mailbox may still be the one HoneyBook is syncing with.
I would first check your email integration settings and confirm that the new email account is the one currently connected. You may need to disconnect the old integration and connect/authenticate the new mailbox rather than only changing the email address associated with your HoneyBook login.
If the new account is already showing as connected correctly, I would reach out to HoneyBook Support through the ? icon. Since you've already tested both addresses and can reproduce the issue, that's really helpful information to give them: emails sent from the old address appear in the project, while emails sent from the new one don't.
I would definitely want to get the integration itself corrected rather than have your team member continue checking two different places. That defeats one of the biggest benefits of having the client communication centralized in HoneyBook!