I don’t think donated organs have ever been put in a cooler before transplantation. That’s probably just a Hollywood trope.
From what I’ve read organs are extracted while a donor is still alive and immediately implanted into recipients.
It still happens. They have special flights even because sometimes a organ is half a country away.
Yes, they do prefer to rip it from your still living body to implant to someone in the same room if possible. Sometimes that's not possible and it ends up in a "cooler".