ESA Animal

I have a doctors note for a ESA animal that was for my dog who now is passed away. I will be moving in April or close to April. I know dog rescues or shelters use to call your landlord to make sure the property allowed animals. If I have a ESA letter, do they still have to contact my future landlord? I don’t know my landlord won’t allow pets, but I have severe anxiety and I swear my brain thinks about and focuses on every existing problem that could come about. Thanks in advance.