Does the USPS check if you were in the military?
Any employer worth working for and certainly a quasi governmental organization such as USPS will check your background. You must tell them about your service or they will find out. Nothing good can come from that. However, you state you got an honorable discharge which is what I would tell them and show a copy of your DD214. I am confused by honorable discharge but no convictions. Not sure what you are trying to say.