CRM: How to keep INBOX from getting cluttered with too many messages w/o losing association with Zoho CRM Contact records

CRM: How to keep INBOX from getting cluttered with too many messages w/o losing association with Zoho CRM Contact records

I have IMAP syncing set up so that I can send/receive emails directly from Zoho CRM but I also use Apple Mail on my laptop, iPhone and iPad for sending mail to / from the same account.

I love that Zoho CRM automatically links the records in my IMAP's INBOX and SENT folders with contact records, no matter where I send the email from - but if I move it out of my INBOX into a different folder, the email no longer shows up in the contact record.

I maintain relationships over many years - I'm visiting a client next week that I've known since 2006!  The work to set up this visit started eighteen months ago.  All of the history behind this is important.  But I don't want to keep eleven years of emails in my inbox!

The Sales Inbox is getting so full that it's getting unworkable.

I need to have some way to get old messages out of my INBOX without losing the connection with the contact record in Zoho CRM. It looks like Zoho doesn't recognize folders created in other IMAP accounts - and they don't recognize the folders that Zoho CRM creates.

I've started to miss some follow-ups because I can't tell whether or not an email that I've read needs further processing. 

What is the best solution to keep my Inbox clean AND keep my contact's email records intact?