time machine can't find a backup

I switched internet companies which required a new email address.  I have a MacAir that I’ve kept with the Mojave OS because Mojave was the last OS where you can retrieve emails via Time Machine.  Well, I deleted my previous email accounts.  But now, I can’t figure out how to get to them because they’re not showing on my Mail tool bar as a valid mail account.  All of them are on the external backup drive but I can’t get at them.  Any insight?  Many thanks.  If I’m dead in the water then I guess I’ll upgrade the MacAir to the latest OS.