Sorry to hear the house hunt has been difficult, it does seem it's increasingly hard to find places allowing animals these days, let alone large ones, so I feel that.
I'm in California and hit the same wall looking for a house, almost everywhere said no animals upfront on the listing. What worked for me was just contacting the landlord anyways, explaining the situation honestly and asking if there was any way to make it work. Some people are willing to have that conversation once it's human to human rather than just a policy. Worst they can say is no!
When I reached out, I prepared some information, like a pet resume with a photo, vet reference, vaccination records and a bit about their temperament, think it goes a long way!
Also offering an increased deposit upfront also helps ease any concerns. A vet letter speaking to their temperament and health is worth getting too, especially given your dog's situation it can make everything feel more real and personal to a hesitant landlord.
Also worth knowing, landlords legally cannot apply breed restrictions or pet fees to Emotional Support Animals. There are legitimate online services where a licensed therapist can evaluate you and provide an ESA letter if appropriate. I'll leave it to you to decide if that route is right for you, but it's an option a lot of people in this situation have used
Hope you find the perfect place soon, good luck!