There are two 'types' of codes you can make, one is a regular redirect to the link you enter (Link QR), the other takes you to a qr.new page that has the information you entered on it (Everything QR). So make sure you have the right type selected and that you've put your link in the link field.

Action item for me though, I need to make that clearer on the site.

I went ahead and changed Link QR to say "Redirect" and replaced "Link to a website" under Everything Qr with "Text," hopefully that clears some confusion.

There's definitely a problem though: when I edit the URL as a Link QR (e.g. changing .com to .co), the little checkmark changes to a spinner then back to a checkmark, but the QR code does not change at all. No error messages in the console other than two script-src CSP errors.

I think that is because the link does not actually send you directly to your website, but routes you through qr.new. If qr.new were to go down, any QR codes created would not work.

This is correct. I have basically combined a link shortener with a QR code.

The change is server side, the QR code pattern shouldn't change. When you click or scan the QR code after you switch types it should behave accordingly.