They are supposed to be tech experts. If that's the best they can do, write directly to the owner of Big Fish and tell him you cannot access 13 years worth of games you've bought from them and maybe he will encourage them to find a workaround. They should be able to do something, even if it's creating a new account for you and transferring all the games to it.
Once that's done, don't ever sign out of BigFish!
"Best not to think about it. I don't want to fall to bits 'cos of excess existential thought."