Just going to cause resentment IMO.
The concept of double checking your own work doesn't really fly in engineering work. Not that everyone shouldn't do it but it doesn't tend to add much value to the end product. It is human nature to overlook your own mistakes.
In HW development, separate people verify and validate everything. The cost of a mistake getting through is too high to just hope it is all good.
The point is that mistakes will always happen. You shouldn't blame staff for being human but the mistakes can't get out the door. If it's that important, a second set of eyes needs to be there IMO.
However since this is a post office, really the important thing for the customer is to get in and out as quick as they can. The major complaint is probably the waiting time. So in one sense the employees are helping customer satisfaction by getting them processed quickly!
At the very least they should know that if something goes wrong because of their error and the business is held liable, they will be fired. If there's no way to know which worker committed the error, maybe that's the first thing that should be fixed.
Bonuses for error-free performance are also a good idea.
I'm all for treating workers well, but it goes both ways: the workers also need to treat the job well.
This world is not ideal. You'll want a caveat: talk to the people who are having trouble, _and make it clear that you're not just here to yell at them_. Your goal isn't to punish, it's to teach. Punishment is useless if you don't tell the person what they did wrong, or if you don't let them fix their mistake.