A way to remember "skriver under" as "sign" is by thinking of it as "writing under" = 'I am not writing under that!', as in I am not writing under the contract on the dotted line.

Works for me anyway :)

In German it means nearly the same: unterschreiben

I am not going to sign that! would be another acceptable translation.

I know that it's not literally correct, but I feel that you would use "this" and "that" interchangeably in English. I can picture someone pushing back from a chair, and gesturing at the paper, saying in disgust, "I'm not signing that!" but otherwise, you would probably turn to the person next to you, and exclaim, "I'm not signing this!"

The combination of D's in the end sounds a bit weird and repetitive. How do I rebuild this sentence to avoid it?

Why does skriver means singing if it's writing?