Cease and desist website?

I have a website called www.bigbirdgrocery.com and a local supermarket chain called Giant Eagle had their attornies send me a cease and desist letter stating that this is trademark infringement because "big bird" is a common trademark name for this grocery chain. They also want me to sign over the domain name and any others like it. We are in the process of changing the name (which although I disagree 150% with Giant Eagle having a trademark on that, I don't have the money to fight a HUGE corporation. My question is: Do I have to "GIVE THEM" my domain names? I bought it, I purchased it. I had NO affiliation with them. I have never worked with them OR can I sell it to them? Thanks so much for your help.