Nine times out of ten this is about where you're entering it. A member's email address lives on the roster entity, not on the booking — if you try to save it directly on a booking, it won't stick.
Where the address belongs
Go to Settings → Roster.
Open the artist.
Go to Members.
Add the member there with their email address, and save.
That's the record Gigwell draws the artist-side booking email from.
Applying it to bookings you already have
Adding a member to the roster does not reach back into bookings that already exist. Those need the member added individually:
Open the booking.
Go to the Change tab.
Expand the arrows beside the artist entity.
Add the member.
Do this for each existing booking that needs the member on it. New bookings pick the member up on their own.
Good to know
This is also the reason itinerary creation sometimes reports "no applicable artist found." The itinerary is looking for a member on the artist entity, and there isn't one yet. Adding the member at Settings → Roster → Artist → Members — and, for bookings that already exist, on the booking's Change tab — clears that message too.
If the address saves but then changes back to an older one on its own, that's a different problem. See the article on an email address that keeps reverting.
If you're trying to give a shared advancing or production inbox visibility rather than an individual, add it as an assistant instead. See the article on copying a shared inbox on every booking.