Screen light off but screen is working.

Just for clarification, you are saying the backlight is out so the screen is dark but you can still make out a picture, right? Or is it that the screen is black but still responds to touch?

That's for my own information; it really doesn't matter which one describes your problem because the fix for either of them is the same; you'll need to replace the screen. Although it is theoretically possible that you could have a failure on the motherboard that's causing the issue you're seeing, in practice that's extremely rare and it's almost always the case that replacing the screen will fix the problem.

In doing my research on this, it appears there are two different models of the A13 that are being sold; there's the SM-A135 that your question is linked to, but there's also a model SM-A137 as well. I'm not sure if there are any significant differences between the two, but you should be aware as it may affect the repair procedure or the ordering process; I have no idea if the same screen will fit both or not.

Here's the repair procedure for the SM-A135.

Réparation Bloc écran Galaxy A13 - Guide gratuit - SOSav.fr

And here's the one for the SM-A137.

Manuals / Samsung Galaxy A13 SM-A137 / Full screen

The first one is in French and the second in Spanish, but generally the Chrome browser does a good job of translating it into whatever language you need so I haven't found it to be a problem.

Good luck; let us know what you find and how your repair goes!