I figured I’d post the answer here as well in case someone else is going to look for the answer.
Why the sale price wasn’t changing completely was because I wasn’t changing al the required fields.
Woocommerce has 3 different fields for keeping the price for every product.
There is the regular price field, sale price and a third one. When I set the sale price, I also had to change the regular price field to be the same as the sale price. Otherwise, Woocommerce did not recognise it.