This is a common problem with 3DS XL consoles. Most of the time when both screens flicker but don't come on there is a problem with the speaker ribbon cable. This is what I check:

1) Remove the back of the console housing and check the speaker ribbon cable to be sure it is seated all the way.

2) Remove the back of the top case (the back of the top screen) and check the speaker ribbon cable there to be sure it is seated and not corroded.

3) Check where the speaker ribbon goes through the hinge. You will probably find a very small tear in the cable where it goes through the hinge and rubs on the metal hinge tube.

If the speaker ribbon cable is seated correctly on both ends then you most likely have a torn ribbon cable and it needs to be replaced.

I hope this helps.