> I can't sing along if it's not close to the original because my brain remembers every tick of the original and wants to match the key of the original as well.

I don't know if this helps at all, but you probably want to learn to sing in a different octave, not a different key. This means shifting your voice up and down from the original in multiples of eight notes. You can sing in a different octave without changing the background music and it'll sound completely normal, and have the same feel as the original song.