You REALLY need to sleep? Fine, I'm just trying to practice but you can watch if you want. I'll run some re-runs at the end for you if you still can't fall asleep, you're good with that right?
To listen while using other apps, switch the player to Picture-in-Picture (PiP) during playback — it keeps playing in a small floating window (the screen stays on).
To listen with the screen fully off, in-browser playback stops by YouTube's design. Open the video in the YouTube app to keep listening where background playback is supported (e.g. with YouTube Premium).
Open in YouTube app