Outlook keeps asking for the password on two of my accounts

Nine times out of ten that's a stale saved password Windows keeps handing to Outlook. Close Outlook, open Credential Manager from the start menu, go to Windows Credentials and delete every entry that mentions those two email addresses (and any that start with MicrosoftOffice). Open Outlook again, type the password once, tick remember. If one of the two is a Gmail or Yahoo account with 2-step verification on, your normal password won't stick at all. Either remove the account and add it back so it signs in through the browser-style login window, or make an app password for it in that account's security settings and use that instead.