Product: Credit card or prepaid card

On or about XX/XX/XXXX I contracted with XXXX and XXXX XXXX XXXX XXXX to transport my furniture from Florida to New Jersey. They were suppose to pick my furniture up on XX/XX/XXXX or XX/XX/XXXX. They came on the XX/XX/XXXX with a truck that was half full ; therefore, they put my furniture in storage and said that they would pick it up Monday or Tuesday. During that week, we couldn't get a straight answer to when they were going to deliver my furniture. So I gave them to the following Saturday and called in a dispute with my Barclay Aadvantage Card that my furniture was not delivered ( services weren't rendered ). 
Sunday as I drove to church, the company called asking why I disputed the charges and if I didn't lift the dispute, I wouldn't get my furniture. So I called the Aadvantage Card dispute department back and the customer service rep. advised me not to lift the dispute because she had seen people not get their furniture ; therefore, because of her being more experienced and representing her company, I took her advice and kept the dispute active. The moving company which I had put a {$1900.00} deposit down initially and then {$2700.00} on XX/XX/XXXX both onto my Barclay Aadvantage card, both amounts were being disputed, told me that I wouldn't get my furniture until the total amount of {$7200.00} was wired to their bank account and then they would release my furniture. 
So, of course, I sent wired the money on XX/XX/XXXX and on a few days after that they delivered my furniture. 
The problem is Barclay Credit dispute department paid the moving company on XX/XX/2019. When I asked them why was the money paid to them and what proof was given by the moving company proving the service was rendered. Their response was that the moving company sent them the contract and that was enough proof that services were rendered. My point to them was that no one from their dispute department called, texted, email or mail me a letter asking whether the services were rendered and plus a electronic signed contract is not proof that services were rendered. Therefore, they paid the company extra money because I had paid them in full. They are holding me responsible to pay the {$4600.00} back to them.First of all, had they called they would had found out that the company only brought two-thirds of my furniture back and my husband and I had to fly back down to Florida at the end of XXXX, rent a truck and drive a thousand miles back to New Jersey.

I am upset and disappointed with the credit card company because I feel that the correct steps were not being done to protect me. It is very unfair they would pay disputed amounts out before verifying if the job was completed and still placed the responsibility on me to pay them. Someone dropped the ball and before verify and confirming whether services were rendered, they paid the company because they sent an electronic signed contract.

I have worked with the dispute department and the Office of the President. Both department are standing together that a signed contract is enough proof that services were rendered. The Extra {$4600.00} is my problem to get back from the company which have gone out of business. The dispute department along with me on the phone tried contacting them and the number is no longer in service. 
I feel they know someone prematurely paid the company but because the company is nowhere to be found, they don't want to take the lose, so they're passing it to me. This is so UNFAIR and I don't have that kind of money to loose. Your attention in righting the wrong done to me by this credit card is appreciated.