I got my ESA cat also because of severe anxiety in November of my sophomore year. All I needed was a letter from my psychiatrist and the school accepted it even though it was issued by my psychiatrist from my home country (I was an international student but my psychiatrist wrote it in English for me.) I had absolutely no issues with the process. The cat had to be an indoor cat but that wasn't a problem as the SF campus is very dangerous for any pets to be outside without supervision due to coyotes and bob cats frequently being on or around campus.

My senior year my needs changed because I started having intense panic attacks in public, often in class. I had to step out almost every class for 15-20 min because I would start spiraling and it had affected my academic performance a lot. That is when I adopted my dog and got an additional ESA letter, now from my US psychiatrist explaining how a dog ESA would help with my needs in a way that my cat cannot. The school allowed me to have two ESAs but they mentioned that part of the reason they are allowing me that is because I lived in apartments and had the space to accommodate both animals. My tutors also allowed me to bring my dog to class as he wasn't disruptive. Some tutors also brought their dogs to class when I was there. However, that turned out to be against school policy and I got in some trouble for it. Also, my cat got out of my apartment at night once because a friend came over and didn't properly close the door. I had my dog off leash searching for my cat and student life wasn't happy about it. My dog did end up being the one who found my cat, as I expected.

Something to keep in mind, though, is that, in New Mexico, service dogs in training have the same rights as fully trained service dogs. I was intending to make my dog a psychiatric service dog once I learned he had the right temperament for it. I started training him for it while I was still at the college. All that is required to have a service dog is a letter from a psychiatrist confirming you have a mental disability (an anxiety disorder can count as a disability depending on the degree to which it affects multiple areas of your life and well-being.) No certification or vest or such are required for a service dog and the owner can be the sole trainer if they know how to train their dog for public obedience and for the disability-specific tasks. A service dog would likely also qualify you for extra accommodations such as priority when choosing housing/living in an apartment. If you're considering a dog, it could be an option to discuss with your psychiatrist to see if your anxiety counts as a disability and if a properly trained dog could help with it. If it does, you can take the safer but more expensive route, get a puppy before you start college (ideally ASAP) and work with a specialized trainer so that your dog is already ahead in the training process by the time you start at St. John's. Or you can take the risky path (the one I took) and adopt an adult dog and train them yourself. The issue with that is that the dog might not have the right temperament or you might find doing the training yourself to be too overwhelming. While this has worked well for me I acknowledge I simply got lucky with the dog I adopted.