Google Blocked Access to My Account Emails, Apps, and Games

I am locked out of my account and cannot access my inbox or my Google Play games and apps. My phone is old, I guess that's the reason.  I can't verify my account and fix the login problem. It's so annoying that even though I know my password,  keep hitting a stupid login wall that won't let me in. I need to upgrade my phone number and keep using it like usual, yet they offer no other solution, can't get out of this loop and use Google Play to download my games and apps, check my email, or keep in touch with my friends. Doing the same things over and over is just getting on my nerves nothing else.