Request by former employer for me to send them money for their mistake.

I was working for a retail company as the store manager, not salaried but as an hourly person at $20 an hour. I informed them I wanted to step down as Store manager and work at a different store within the same company. They hired a new store manager. On the day of the transition, his first day at my old store and my first day at the other store they had not made correct payroll changes in order to transfer me. He had been correctly transferred. I realized they were still listing me as a store manager and paying me $20 and hour, not $14 as I expected. For the next three days I notified my superior every day that I had not been transferred yet. On the third day he told me the head of corporate hr was informed by him and is on it. I worked only three days. When I came back the following Sunday to start my second week the same thing happened, I had not been transferred into the new store's database. I again informed the store manager and he told me HR is aware. This is a publicly traded company with a NASDAQ listing and 285 stores and five thousand employees. After the beginning of the third week I told the manager again that I was still not in the correct system and probably was going to be overpaid. He said HR knows what they are doing, he could not do anything about it. On that day for other personal reasons I gave my two week notice. I left the company two weeks later and for two pay periods,( four weeks) they overpaid me. Now a month later they are asking me to sign a letter and give them back the net difference. My questions are Do i have to legally give this back to them? What law compels me to do this? I notified them and did not fraudulently change hours or work extra shifts. Also what about the payroll taxes deducted at the higher wage rate? Is that accounted for? Can I get that back from the FEDS? They gave me no spreadsheet accounting, just a one paragraph letter saying they overpaid me and to sign the letter saying i will pay them back $650. I made no mistake, I worked hard for that money. I spent it on rent.