HOW to disable this ME.com without affecting my existing emil with MAC.com ?

Ask Apple if they can do anything about those old accounts.

Refer to this page for Apple Support features ➔ Contact - Official Apple Support

Select from the presented options until you find a solution for your issue, or see if there is a chat or phone call contact method offered lower on the page under “Get more help” (you sometimes have to narrow down the options multiple times before this is shown). If you do not see your issue, keep experimenting with any series of selections until you reach one that offers a chat session or a telephone call and get the representative to redirect you.

This support article has various country telephone numbers for contacting Apple for support and service ➔ Contact Apple Support - Apple Support For some countries look under “Other” listing at the end of the list. Not every country has telephone support, and phone numbers may only work when calling from the same country.