Hi Jacob,
This should work with a smart campaign:
1. Smart List: Trigger - Score is changed
Select your score field and add a the New Score constraint and se the value that you are looking for. You'll probably want to use the is grater than option.
2. Flow - Change data value
Attribute :Select the custom fiels that you created
New Value : {{system.date}}
Is that any different to what you were trying?