Hey there @Callie38318 - great question! New User Guide

For some context, with Calendly, there are two types of notifications: Email Confirmations & Calendar invitations.

1. With Email Confirmations, Calendly sends the invitee an email notification from notifications@calendly.com and has a reply-to address that's associated with your Calendly login email. You can modify this on Account Settings page.
2. With Calendar Invitations, your connected calendar sends the invitee a notification of the booked meeting which has a reply-to address that's associated with your Add to calendar email on your Calendar Connection page.

Note: Calendar invitations are not compatible with iCloud Calendar or the Outlook Plugin (for Windows users). Users with either of these calendar connections set up as their "add-to calendar" on their Calendar Connection page will be defaulted to Email Confirmations. 

With this said, if you changed your login email address that will only impact the email address invitees see if you are using email confirmations. If you are using calendar invitations, you will need to update your “add to calendar” because that email address is what is associated with those calendar invitations that invitees see.

Your own booking confirmations will always be sent to the login email address on file and bookings will be added to your add to calendar. These do not need to match! 

I hope this helps!