What about www.mywebsite.com?
If you have your own website domain name like www.woofysrescue.org and you want to use that URL for your PAWsite, if your domain name service provider offers "domain forwarding" you can have your domain (like www.woofysrescue.org) setup to automatically forward to your PAWsite public link (like http://www.adoptapet.com/woofys-rescue).
So someone would type www.woofysrescue.org and they will be forwarded to http://www.adoptapet.com/woofys-rescue.
Please contact your domain name service about domain forwarding. Adopt-a-Pet.com does not offer domain name services.