These days when I say I need to check my calendar people are instantly disappointed, everyone knows it’s beyond full.
And I can memorably remember being screamed at extensively (by a higher up outside of my chain) in a meeting 20 years ago, for asking what needed to be deprioritized.
I mean I use these all, but if people are going to be emotional about a no, they will see right through these, and you will have the same problems.