“I can’t this week” is enough information.

A refusal does not need evidence attached to it. The urge to explain comes from a suspicion that no is only allowed once it has been proven, so you offer a reason and the reason gets debated. Short answers are much harder to argue with than long ones. Fewer words also carry less apology inside them.

Try declining something this week in under ten words, with no story and no follow-up. If somebody presses, repeat the same sentence instead of building a better case.