RK, thanks a lot for answering my question! I had thought that it was PirateShip that was sending me to the old API, but it was actually a rewrite function in my Nginx configuration that was rewriting PirateShip’s request.
I tried for quite awhile yesterday to figure out the correct Nginx configuration to allow access to the WordPress and WooCommerce API, but all my attempts failed. I ended up reinstalling my site using Apache instead, which has much better out of the box support for WordPress’s API.
Hopefully this is helpful for someone in the future!