You could also look at Bitwarden. I'm not affiliated, but it's an amazing password manager (desktop app, mobile apps, browser extensions). Even better, it's fully open source; you can even run your own server to keep out a third-party being your (encrypted) sync server. I've been using it for a couple of years now, and have never had a single problem. Free for personal use of almost everything… an upgrade to include the two-factor auth you're looking for is a very cheap $10/year.

Yep. If you have the desktop app installed, look for Two-Factor Authentication under Watchtower and it'll tell you all the sites that can use it. It's easy to set up too, it can scan a QR code on your screen. Some people believe that it's more secure to not mix your password manager with your 2FA though, since it's not "true 2FA".