Apple ID password doesn't match (Macbook Air)

I want to upgrade my MacBook Air (from early 2014), the current software is OS X 10.9.5 and I want to upgrade it to OS Mojave. However, some apps on my Macbook won't allow me to log in - for example iCloud and AppStore. I have successfully reset my Apple-ID password through appleid.apple.com, and I am connected through my iPhone for the two-step-authenticication. My AppleID is correct. And I'm also connected to iCloud on my iPhone.

Why doesn't my Apple ID-password match with iCloud, iTunes or AppStore on my MacBook? It seems that somehow it doesn't connect to when I do any changes in my account. I've also tried to log out of the apps, restart the computer and log in with my apple-ID and new password. It doesn't change anything. It asks me to fill in my password as well as the numbers showing on my iPhone (authenticication), but it doesn't give any alternative to where I can write the digits. I do not have any app-specific passwords (I've also checked the info on my account). I have tried to log on with my new password without any luck, and then double checking if I can still log on to my Account (and yes I could) - so the password is correct.

What can I do to connect my Apple-ID so that I can access AppStore and download the software?

Thankful for any help!