Last night, we called the USDA office in Richmond and spoke to a very helpful woman. We made some arrangements and after all scanning/sending/stamping/returning/printing processes were done (the same vet re-signed and re-endorsed every page), and they carbon-something-ed the seal so the raised edges were apparent in the scan, we printed the end result and brought them to the vet along with the cats.
The vet was kind enough to say that he would release them from quarantine after their checkup tomorrow, but we will need to get the originals sent over ASAP. We can deal with that.
Huge sigh of relief!
Thank you everyone for your concern and sympathy. I'll share something more amusing next time, I promise. Like the aquarium in Tokyo or Nekobukuro, the cat petting zoo.