I do not know why an agency would advise you to lie to immigration. Lying is never a good idea. Whatever you do, you can never disclose to immigration now that you were common law partners at the time your partner landed as a PR because that would mean that your partner committed misrepresentation when they landed and they could lose their PR if that comes out.
The correct procedure would have been to include each other as common law partners on your applications and the first one to be approved, you would both have gotten your PR and cancelled the 2nd application.
The correct procedure would have been to include each other as common law partners on your applications and the first one to be approved, you would both have gotten your PR and cancelled the 2nd application.