Jem, you need to be sure the sink you ARE NOT plunging, has it’s drain covered tightly when you plunge the other drain.
I don’t know if I’m making myself clear.
You are using a plunger, aren’t you?
I reread your post and it appears you aren’t using the toilet plunger on it. Give it a try, being sure the drain you aren’t plunging is closed tightly.
I can think of one, maybe 2 times in the last 40 years that I got assistance while attempting to unclog a drain.
Just because they’re born knowing this stuff doesn’t mean they are motivated to remove themselves from a Lazyboy to assist us.