Apple Mail

Hello - I thought I had clicked on Tech support, but here I am in a discussion space?

Hopefully someone can help - the last few weeks has been a nightmare with my hotmail on Apple mail. 

Am I going to need to change all the emails to outlook officially, and let everyone know (hundreds and hundreds???)?? Or does hotmail stay while the magic works in the background as outlook?

I've tried resolving this at least 6 times with Apple Support (which actually exists, unlike MS.......), but I wonder if I need to fully change my email handle before it will quit giving me grief and work with the Apple Mail??

Thanks