One more update:

I contacted the Austin Tenants' Council. They said that 1) because the apartment complex gave more than a 30 day notice of the change of policy and 2) they are not trying to enforce it under my previous contract with the previous property management - they are within their right to establish these rules.

It goes without saying: "Just because they have the right to do something doesn't make it the right thing to do."

They also said that I am within my right to post a flyer at the bulletin board at the mailboxes. They said to be careful that they might remove it, so it might be the best idea to do a petition as I had planned by having individuals contact me through an anonymous email.

And yes, I will definitely move instead of even considering declawing my cat!