So I did some math and ordered my plane ticket while my passport was still processing. Now it turns out that my passport won't be here in time for my travel date. Is there anything else I can provide at the airport to show I have a verified passport or something to get through?
Unfortunately, you'll need a (real, physical) passport to enter Poland, no ifs or buts
You can request an expedited passport by mail for an extra fee, or there are 28 passport centers across the USA where you can get same day processing for an even bigger fee. Since you've already mailed in an application, you may want to call them on the phone (1-877-487-2778 TDD/TTY: 1-888-874-7793) to see if you can expedite your current application.
More info can be found on the State Department website
The update to this T.SE question notes the poster's experience with third-party expediter service.
The US Department of State website says:
If you are traveling within 2 weeks or need a passport for a foreign visa within 4 weeks and: Have already applied, please contact our National Passport Information Center (1-877-487-2778) to speak with a Customer Service Representative when you are 7 days out from your travel date.
So if you're within a week, you may be able to pay to have the passport mailed to you more quickly for instance.