Complaint sub-issue: Information belongs to someone else

I have been working on a mortgage with a lender since early XXXX to purchase a house. Equifax reported a freeze on my file to the lender, who then instructed me to contact Equifax to remove the freeze so that the lender could access my file. After speaking to a customer care representative at Equifax, I learned that they were unable to locate a credit file with my information at all. At this point, she instructed me to fax a copy of my driver 's license and Social Security Card to Equifax via the fax number XXXX. I sent this fax the following day and received a XXXX Confirmation Report saying that the fax was successfully transmitted. I received a letter in the mail the following week dated XX/XX/2020 saying that they were unable to locate a credit file with my information, which is the problem that led to my sending the identification information in the first place. So here I was back at the beginning of the problem. I contacted Equifax again and was assured that the problem would be handled and asked to fax my information again. I then faxed the requested copy of my driver 's license and Social Security Card again on XX/XX/XXXX. I received a confirmation just as I did with the first fax, and waited for a response once more. A week later, I received the EXACT SAME letter as the first time dated for XX/XX/XXXX this time. This continued until I had faxed my information over 4 separate times, the latest being on XX/XX/XXXX. I have yet to receive any response from the fax sent on XX/XX/XXXX, and spoke to an Equifax agent on the XXXX who told me that he had started to resolve the issue and would be uploading my documents to my credit file and that it should all be fixed in time for my home closing on XX/XX/XXXX. He even provided me with a confirmation number. I have spoken to at least 3 different Equifax employees since then who have told me that they have no record of the confirmation number and that the agent did not in fact do anything to help my situation. 
After speaking with a supervisor named XXXX at Equifax today, who claimed to work in both Customer Care and Disputes, I was told that they had no record of receiving any of my paperwork. So that tells me that they have somehow lost or misplaced at least FOUR copies of my driver 's license and Social Security Card, which puts my personal information at very high risk of future identity theft. I have been incredibly disappointed with the level of customer service that Equifax has provided. I have already closed on the sale of my home, and can not close on the purchase of my new home until this issue is resolved, which has left me HOMELESS and Equifax couldn't care less. They have someone named " XXXX '' associated with MY SOCIAL SECURITY NUMBER and they have made zero effort to fix it. I find it absolutely absurd that they can't be bothered to resolve this issue and they have lost all of my paperwork and yet somehow we are supposed to trust them with our personal information and trust their judgement when applying for our mortgage. The urgency of my situation means absolutely nothing to them from what I have experienced. Also, I was informed today that they are no longer accepting faxes, so I had to mail my paperwork this time, which takes even longer than the fax method.