After speaking with my mother, she suggested I call the Humane Society of North Texas to see what they would charge, if anything, to put Millie to sleep. After searching their website I found this: K911 Emergency Emergency Rescue. Here's what their website says: "HSNT is the only local humane agency that attempts to rescue and alleviate the suffering of sick or injured animals on a 24-hour basis. The phone number for K911 is 817-891-5911. For uninjured strays, please call your local animal control. Due to limited resources, HSNT is unable to provide veterinary services. Humane euthanasia is offered when needed to alleviate suffering."
I called the number and spoke to someone who was very kind. He told me that I could bring my dog in and they would euthanize her at no charge. But since they are a nonprofit organization, they ask for a donation - whatever you can afford. Such a load off my mind. What an awesome organization that offers help to suffering animals and their owners.
I didn't wind up needing their services. Our Millie passed away naturally and peacefully at home yesterday evening.
Thank you, Humane Society of North Texas, for all that you do day in and day out.