Hangouts operates independently of Chat, and has no impact whatsoever on whether or not Chat works with your account. The only issue is that, when using Gmail, the "call phone" feature will either use Chat,
or Hangouts, but not both. You need to at least temporarily revert your Gmail account back to using Chat, as I documented in my instructions (did you actually read through and follow those instructions completely?).
Please take a screenshot of this web page:
https://www.google.com/voice#phonesObscure (block out) your private information (email address and all but the last 4 digits of the phone numbers), and then click "Additional options..." below, to attach the screenshot with your reply.