I've recently changed the main email address I use. I put the new address into LinkedIn, confirmed it, and made it primary. But LinkedIn still sends me all its email on the address that I had first joined it with (which is still active, so I don't want to remove it.) How do I get LinkedIn's messages to come to my new primary email address?
Unfortunately, you seem to have followed the correct process.
I assume that you can distinguish the emails. So it is not possible that you have a forward set-up outside of LinkedIn.
Otherwise, as you say you have multiple active emails, I would suggest changing the primary email back to your original one (that still gets the emails). Wait for "1 day" then try again. ( Wait 1 day if this is not urgent, otherwise an hour should be plenty for back-end systems... assuming that there is actually some kind of lag )
Are you having this problem 24 hours after you changed your primary email address?
According to the LinkedIn suport site, changes to your primary email address should take effect within 24 hours.