Trying to backup my iPhone 6s with an unresponsive screen

A month or so ago I dropped my iPhone 6s and the screen stopped working. When I turn the screen on, I can interact with the screen for maybe half a second. Then I have to turn the screen back off and on again to interact for another half second. After the half second is up, the screen starts to get these pink links and spots that cover the screen and the screen becomes unresponsive. I decided since it was already an old phone it was better to just get a new iPhone rather than having to send this one into Apple and wait weeks for it to be fixed (on top of the fact that the fix itself would probably cost more than the phone is worth).

I just bought the iPhone 12 Pro and will be picking it up on October 24. However, the last time my iPhone backed up was October 12 (which it did automatically). While I haven’t been using the phone recently (obviously), I’ve still received texts and other notifications that I would like to keep. I tried to make it backup automatically by plugging it in and waiting overnight, but when I check iCloud it still says the last backup was Oct 12. I think it has stopped backing up automatically every 24 hours because when I turn my phone on and sign in using my fingerprint, it asks me to enter my Apple ID password. Obviously I can’t type it in, because I can’t work with the screen for longer than half a second. If I rapidly click cancel, the next time I turn the screen on it pops back up. I tried using a voiceover and my laptop keyboard to enter the password manually and navigate around to make the phone backup, but no dice (unless I’m doing it wrong?). I saw a YouTube video where the guy seemed to be doing this method successfully but it does not seem to be working for me (https://www.youtube.com/watch?v=dciP0geH624).

Am I going wrong somewhere? Or any ideas as to how to backup my iPhone?

Thank you for your time pals 🙂