I have shipped over 3000 items through Etsy, I print out the mailing labels when I'm done making the product and stick them in a bag by my mailbox for my postal carrier. Sometimes they're printed out after she's already picked up the mail, so they my mail carrier picks them up the following business day (If I print on Saturday, that's Monday). Sometimes I have a lazy mail carrier and they elect not to take the packages waiting for them, so they can sit for an extra day. - then I usually take them in to my local PO, still not a problem.
I have not once, ever, had a challenge. Not ever, never! In fact, before reading this thread it never once occurred to me that the date 'shipped' on the package meant anything to the USPS.
I would encourage you to send your package with your postage paid in full.