How do I fix Gmail going offline on Mac Mail after changing password?

I have looked up past questions on this and they all seem to be unresolved. I have done all the troubleshooting suggested - nothing works. My regular Gmail works if I go online through my Google account but it won't receive or send any emails on my Mac. Just has a red dot that says "off line." Further, when I do try to re-set it up on my Mac mail, it says "this mail account already exists." However, it does not provide any place to put in a password under account settings for Gmail, and it also will not allow me to manually make it go back online. I have quit mail and restarted it numerous times, also restarted computer, and yes, my operating system is fully up to date. Any other suggestions?

[Re-Titled by Moderator]