People who send me emails get a"mailboc full" response, yet I've cleared it down to a minimum

mailbox full

Generally all an email client does is give a synced version of what is on the server. But if it is setup as POP3, then deleting messages in eM Client might not delete them from the server.

If you go to Menu > Tools > Accounts, does your email account have POP/SMTP tabs, or are they IMAP/SMTP?