Abandoning Yahoo mail for ..?

I will not stay with yahoo mail for much longer as they are really pushing me to "upgrade" to the new email version where you need to agree to have your email scanned for targeted advertising and for the three letter agency
They invalidated the Search Inbox function and present me with the UPGRADE splash screen instead in last days

I can set up IMAP mail and pull my 10+ years of email into Mail on my MB Air and then close the account
Then, I would like to use my own domain email solution with 1&1 with the same email database, is this possible by just changing the mail server info ?
(E.g. I would like go to mail Inbox and reply to an old email from my new email address).
Of course, now I need to think about mail backup etc.
Any recommendations how to handle this unevitable transition ?

I would not touch gmail for the identical privacy reason. I switched to startpage as seach engine not to give google any more info than needed.

Mike, what you describe is exactly the way I am going. The $24/year price for the meail package is worth it.
I need to try whether what I tried to describe and you rephrased, will be working. Will Mail (set for my new email address) protest if i will try to reply to an old message sent to the previous email address ?

From reading macintouch, there is no worthy replacement for Eudora and people gravitate to Mail after a while. Also need to find out how to reliably back-up the mail, Time machine or separately some folder..