Q: After initial login I still get prompts to login to mail safari, etc

Hey there Mary! 

One password to login is usually more than enough for me.  We can certainly check out what's going on here.  

If you haven't, we'll try a couple things to find out more.  Let's start with restarting the computer in safe mode to see if the logins still show up.  Then restart normally and try the same with a test user. 

Try safe mode if your Mac doesn't finish starting up - Apple Support

How to test an issue in another user account on your Mac - Apple Support

Let us know how those go and we can see what's next.

Take care.