You're doing something that is hard without a stutter; give yourself a lot of credit and feel confident that you're doing right by your mother. Advertise that you have a stutter when talking with a new person involved in your mom's care, just get it out of the way at the start of the interaction. It doesn't have to be long, just a short, "Hi my name is \[ \], and I stutter." They all work in healthcare, they'll understand. After its out in the open, you can focus on what you're trying to say and helping your mom--don't worry about anything else.