Thanks for your reply and your continued help. I disabled Cloudflare and changed the nameservers back to my original host's. USPS shipping is still not showing up as an option. I've confirmed that the page is being served by my host and no longer Cloudflare.
When I first view the cart, the form data sent with the POST (request URL "Request URL:.../store/index.php?rt=r/checkout/cart/change_zone_get_shipping_methods") sent by the shipping estimator is "country_id=223&zone_id=3663&postcode=", which makes sense, since no zip code was inputted yet. The response is "{"default_usps":{"title":"United States Postal Service","quote":[],"sort_order":"0","error":"Missing value for ZipDestination."},"selectbox":""}", which also makes sense.
When I input the zip code in the shipping calculator, the URL request is the same, and the form data includes the zip code. However, the response (in the browser's console) is blank.
How do I check the XML sent to the USPS's API and the response?
ETA: My workaround for this is to set up weight-based shipping, which I can easily do because all of my items will ship USPS media mail, which is solely a weight-based shipping method. At this point, I'm trying to get to the bottom of this for the sake of others who might be facing a similar problem. Thank you!