I’d be looking at a used 2070/Super or 3060/Ti in your situation.
If you’re on Windows, consider that you have a year of support left with Windows 10. You can buy further security updates from Microsoft, from 0patch, or with an “unsupported” Windows 11 upgrade. There’s also Linux if the games you want to play are supported :)
With that in mind, you can consider the alternative of limping along with what you’ve got and saving for the next year to do a more substantial upgrade / overhaul.
After you power the system on, does pressing the capslock key cause the corresponding light on the keyboard to change states?
Once the system is powered on, does the monitor and its backlight stay on? Or does it fall back to a “no signal” mode?