Can't log in with Apple ID on OS X Lion. Verification code isn't working.

Hello, recently I dug out my MacBook pro mid 2012 and reset it completely. I reinstalled OS X Lion on it through internet recovery and everything is working fine again.. kinda.

Even though I'm connected to the internet and apple.com does load when I launch Safari, every other website refuses to load and the App Store barely functions where it won't even load any of the images, just the text on the homepage, but nothing else.

I assume this is because OS X Lion is so old that it just has a hard time to keep up. No worries, I wanted to upgrade anyway.

But here's the main issue; I can't log in to my Apple ID to download a new macOS version.

When I enter my credentials, it gives me a pop up saying that I need to enter the 6 digit verification number after my password. I get the code on my phone, enter the 6 digits after my password as instructed, nothing else, and try to log in again, but nothing works. Then it just says "Unrecognized Apple ID or password" and sends me another code to my phone.

I've tried this nearly 20 times over the last couple of hours and nothing has changed.

I'm receiving the code, so I'm definitely connected to the internet and using the right credentials, but why can't I log in?

Side question; Is there another way to upgrade to, let's say, macOS Sierra without using the app store?