OK, here are the Regulations
https://pe.usps.com/Archive/PDF/DMMArchive20041209/S020.pdf
2.0 CASHING MONEY ORDERS
Validity and Value
2.1
Domestic money orders are paid regardless of the time passed since their issue.
Money orders are not interest-bearing instruments; they are paid only in the exact amount imprinted up to the authorized maximum amount. USPS records serve as the basis for adjudicating claims for payment of money orders.
Redemption
2.2
All U.S. money orders, including military, may be cashed at any U.S. post office or bank. Subject to funds availability, money orders may be cashed by rural carriers.
Identification
2.3
When presenting a money order for payment, the customer seeking payment must sign in the presence of a USPS employee. If the customer is not known to the employee, suitable identification can be required. The USPS may refuse payment on any money order when the identity of the customer seeking payment is not proved to the satisfaction of the employee.
No stamp is required at the time of paying (you can check the rest of the regulation if you like, but a word search does not reveal "stamp" anywhere else except in reference to "stamped signatures" and "postage stamps").
Acceptable Signature
2.4
The paying post office may accept any signature of the payee, purchaser, or
endorsee that is not different from the name shown on the money order, subject to
these conditions:
b. All money orders payable to a business firm, an organization, society,
institution, or government agency must be signed in the name of the
organization by an authorized representative (who must also sign with his or
her own name and organizational title). Evidence of the representative’s
authority may be required.
e. A stamped signature is an acceptable endorsement on a money order drawn
in favor of a firm, corporation, association, society, or individual, when the
money order is presented to a bank for payment. A post office accepts a
stamped signature only if an agreement is on file specifying the individual
responsible for the correctness of such payments.
This makes perfect sense. When the Payee is a business, they would not take cheques and money orders to the post office or bank and cash them and then bank the cash, they would bank them directly into their own bank account. They are listed on a deposit slip and bundled with it. The bank would then clear the whole bundle as a batch, which is where the key punch operator would come in (remember, this is 1963 and MICR encoding of cheques was only just being introduced). Klein's Sporting Goods was a fairly large business, advertising all over the USA, they would likely have banked hundreds of cheques and money orders each day.
should have plenty to keep you 
