Would they sell the swaps back to the banks that had originally issued them? If not, what party would the original buyers be selling their swaps to?

Digging deeper (please excuse my lack of knowledge in this space), if the credit default swap is essentially a contract between the buyer and seller in which: (a) the buyer pays the initial amount plus the cover each month, and (b) the seller promises to fully payout in case of the underlying default, then if the second condition of default is recognized on the part of the seller, shouldn't the payout be the completion of the contract/transaction? Why does the buyer need to "sell" the swaps in the first place? I'm closely drawing the analogy to the one presented in the movie about fire insurance: if someone purchases fire insurance and a fire occurs in his home due to no fault of his own, the insurance company would simply reimburse him for his losses; he wouldn't have to "sell" his policy.

So this gets back to my original question of who the swaps were sold to once they wanted to recognize their profits.

If they sold it back to the banks, would the transaction work in this manner: (1) Hedge funds "sell" the original swaps they purchased back to the bank. The bank gives the high pay-outs to the hedge funds as per the original contract. The credit default swap is no longer in existence because both parties have fulfilled their obligation.

Alternatively, if Baum, Burry and J&C sold the original swaps to 3rd party investors, would the transaction work as follows: (2) J&C sells their swaps to 3rd party investors. The 3rd party investors are waiting for the value of the underlying bonds to drop even further. The 3rd party investors wait for an opportunistic time to finally receive payout from the banks.