Turns out it wasn't the vet that called Alexa but a wonderful lady from the shelter. Earlier today Alexa called them to let them know about Dewey just so they could be aware. The lady from the shelter called the vet and spoke with him. She told Alexa that there is no way to be 100% sure on distemper without doing an autotopsy. She asked if we would be willing to take him to another vet for a second opinion. This was something both Alexa and I had discussed earlier. We still couldn't bring ourselves to call and make an appoitment for him to be put to sleep. The shelter is going to pay for us to see a different vet. We are still waiting for her to call us back and let us know when we will be taking him and where.