I'm thinking of lying to avoid covering shifts on my days off. The person I care for at work is completely disabled
Why do you want to avoid covering shifts?
I'm not sure I want to work on my days off.
That seems logical. It is totally understandable not to want to work on your days off.
Damn you, never listen to my thinking.
But I am listening to your thinking. I agree that it's okay to not want to work on your days off. Maybe you should talk to one of the higher ups and explain your dilemma to them?