I decided to look at the postal regs a bit closer, since I was surprised that the PO could just give them to somebody else. Thank you for pointing that out, it was interesting!
For anybody who might be interested, I am copying this from the USPS site:
524 Disposal
Any mailpiece containing live animals that cannot be delivered to the addressee or returned to the sender within 72 hours (for live, dayold poultry) or within the delivery period marked on the parcel (for other mailable animals) must be disposed of immediately. For safety reasons, dispose of mailpieces not marked with a delivery period when it reasonably appears the articles cannot be delivered or returned in a viable condition. See POM 691.52 for disposal information. Also see 742.3 for the disposition of mailpieces that are refused by air carriers.
When referencing live animals, saying that they must be 'disposed of' made me cringe, so I found what it said in POM 691.52:
691.522 Noninjurious Items
Dispose of other perishable mail, drugs, and cosmetics as follows:
a. Perishable Items. The postmaster must sell immediately all salable
perishable matter that cannot be forwarded or returned before spoiling,
day-old poultry that cannot be delivered or returned within 72 hours
after hatching, and other animals that cannot be returned to the sender
alive. The addressee and postal employees may not purchase any
such item from the postmaster. The postmaster must send the
proceeds of the sale, less a 25 percent commission (but not less than 1
dollar) and the proper money order fee and postage to the mailer by
postal money order, with an explanation on PS Form 3820, Sale or
Destruction of Perishable Mail, of the action taken. Deliver all
perishable articles or animals that cannot be sold to any public or
charitable organization that will accept them. Also try delivering live
animals to a local humane society or animal shelter that will take them.