Since there is no official response for more than two weeks, I decided to correct the situation myself.
After the second reinstallation of the extension and server reboot, I was able to update the authorization parameters from DO.
Yes, because of this, I had problems with the Let's Encrypt certificate renewal. The extension could not add a TXT record to DNS for verification.
After that, I discovered that DigitalOcean DNS Extension gives Server Error 500