Household Bank Platinum Mastercard Charges to my credit card without providing verification of charges San Jose, California

I opened up a credit card with Household Bank in 2003.  The 1st charge on it was for 2500. to pay of a vehicle. Other various charges were made by me throughout the years and were put on my card, and I faithfully sent in the requiredminimum payment and more every month for the past 8 yrs. Because I am an honest person, it never occurred to me that Household Bank would add charges onto my card on their own. 

I noticed a balance transfer on my card 5 or 6 years ago, questioned them on it and never really got anywhere. A few months ago, I was looking at my monthly statement, and noticed 3 balance transfers were put on my card.  One for around $600. and two for between $1500 - $1700 each.

I called them about the 3 balance transfers and they said I would have to call another #.  I called another # and they said I would have to write a letter.  I wrote the letter, and they can only "say" the dates that the balance transfers were made, but, cannot or will not furnish any other verification of the balance transfers to my card.  They said 1 was made in 2003 and 2 (6 days apart from each other) were made in the month of July 2006.

I asked several times to several different people, who made the balance transfers, and, what credit cards were they transferred from, onto my card. No one seems to know, or else, they are withholding this information from me.  I don't know where to go with this from here. 

I received a letter from them saying, "Unfortunately, because we only maintain records for 4 years, we are unable to supply you with all the information requested.  We regret any inconvenience caused by this matter".

I Dont Even Know How To Do A Balance Transfer, I Have Never Done One!!  I told this to them. 
So, here I sit with around $3000.00 added to my credit card, and no one seems to know the Who, What, Why, Whatever information regarding these balance transfers.

I have called and spoke to several different people (including department managers) and no one can give me any information or verification of these charges onto my credit card, and yet, they say I owe them.

My credit card is probably paid off a couple of times over. The people that I have spoke to just keep saying that they don't know who made the balance transfers, they don't know what credit cards were transferred onto my credit card, they only know the dates that the transfers were done.

I don't know who to talk to from here.  I am Far From Rich, I live check to check, I only have 1 other credit card in my name, and have paid it off a few times (but currently owe on it), and have never done a balance transfer in my life.  The last person I spoke with said that a balance transfer is done on the courtesy checks they mail out every month to card holders.  I have NEVER used one of those checks.  If I write out a check, it comes from my checking account, not from thin air.

If anyone can assist me with this, I would really REALLY appreciate it.