I don't know about the employment practices of the Postal Service (and I don't know if any other criminal defense attorney would), so I can't tell you how long their pre-screening process might take.
California Labor Code 432.7 says an employer can't ask about any arrest that did not result in a conviction; however, an employer CAN ask about an open case where you are on bail or released on your own own recognizance. In a deferred entry of judgment (DEJ) arrangement, the defendant enters a guilty plea, but the judge continues the case for sentencing instead of accepting the plea. If the defendant complies with the terms of DEJ and doesn't commit any new crimes, the case is dismissed. California law does not consider you convicted while on DEJ, but you would be released on your own recognizance pending sentencing of dismissal of DEJ, so an employer could ask about it.
However, since the US Postal Service is a Federal agency, they can play by a different set of rules and may view your DEJ differently. For instance, in cases where a person is not a US Citizen, immigration officials consider the person convicted when they enter a plea, so a person on DEJ becomes deportable. If they complete DEJ, immigration officials no longer consider the person to have been convicted and they can no longer be deported on those grounds.
Without reviewing the actual application, it's hard to say whether you should have disclosed this case to the USPS. It might have been better to wait until the case is dismissed in September before filing the application.