People who work in animal rescue make minimum wage unless they are either a vet, vet tech or have a professional skill that the organization needs such as Executive Director, accountant, etc.
You cannot be a veterinarian, veterinary assistant or veterinary technician if you are squeamish. Yes, many people do get over being squeamish.
You may wish to consider majoring in something like public administration plus business or human resources if you want to be part of the administrative team that runs a non-profit. But if you want to be a pet adoption counselor or work the incoming desk or clean/care for the animals, you'll be making minimum wage.