What is the natural way to say it?
"while I'm away" is more natural than "while I will be away". However, I think, it can be skipped, since you both know the circumstances, and adding it may sound formal and make the sentence lengthy.
(Being spontaneous makes what you say 'natural'. Say it aloud a couple of times and you'll know how to do this. I did.)
Here're some alternatives, just to help :
- "Thanks very much for your offer to care for my cat".
- "I can't tell you how much I appreciate your offer to help".
- "Please know I appreciate your timely offer to help".
- "I want you to know I'm grateful for your offer to care for my cat".
And, you can always add a little extra by saying: "You will receive a lot of cat-love". Or "Don't steal his (or her) heart while I'm away".