macOS Mojave App Store won't sign in.

Never mind, just figured out an easier way to resolve this:

1. Spotlight > System Information.
2. In the lefthand column, locate Software > Applications.
3. Find "App Store" in the list. If there are two versions, click the older one and identify its location.
4. Navigate to that location in Finder and confirm (via right click > Get Info) that it's the older version. Delete it.

Voila!

"but now when I go to sign in, the login box disappears like the sign in was successful, but it doesn't actually ever sign me in."

Anti virus software or net monitoring apps installed?

I suggested a restart since you hadn’t mentioned it in your original post.

I wonder when they are going to fix this. I have been searching for a solution (on and off) for the last two weeks until I found your post. Having the exact same issue—logged out, tried to login, took my credentials, but not actually logging me in.

Before I logged out, I was getting the error message: "Failure to connect" and I tried proxies, DNS flushing, restarting, etc., etc. but eventually tried the logout/login sequence. Since then, I've been logged out. Interestingly, when I put in the wrong password, it will give me an error and give me a chance to retry. So, clearly there is not an authentication error. Something else must be either missing or not connecting.

See if restarting your Mac helps.

I did the same thing, it worked, but when you shutdown, it strips everything you did that BocaBoy recommends to do and you are back at square one.