I am sure there are two different issues here.
The 'normal' overpayment might be LOA for a short time when you ceased to be eligible or a get-you-home payment that is reassessed as ineligible. In the latter case by sgt had been claiming as his journey home was over 200 miles: he didn't drive and had to go via Newcastle. Our helpful scribblies decided that the direct journey was less than 200 miles so they reclaimed the overpayments in one big chunk until we managed to point out that he and his familiy had some minor living expenses, such as food. The repayments were adjusted.
As others said, you have £6000. Bit of a mess but best sorted out 'in-year' to avoid tax problems etc. Put it in a high interest ISA paying monthly and pay back as slowly as you can.
I was paid for 3 months after I retired and made some £60 tax free on the deal.