Have a delay script: This gives you time. Leave no wiggle room. Keep answers short and simple.
“Thank you for asking. I just don't have any room in my schedule.”
“My plate is full but thank you for thinking of me.”
Set your own rules: Decide ahead of time what is an automatic pass.
Written replies: Text or email may be easier that an in-person response.
Reframing: Whose feelings are really getting hurt when you overcommit or get overwhelmed?
You? Your family or pets?