If I am now living common law, can I continue to claim single status or do I have to claim common law status?

Canada Revenue would like you to declare your marital status based on the facts of your situation, as follows;

Common-law partner
This applies to a person who is not your spouse, with whom you are living in a conjugal relationship, and to whom at least one of the following situations applies. He or she:

1. has been living with you in a conjugal relationship, and this current relationship has lasted at least 12 continuous months;

   Note
   In this definition, 12 continuous months includes any period you were separated for less than 90 days because of a breakdown in the relationship.

2. is the parent of your child by birth or adoption; or
3. has custody and control of your child (or had custody and control immediately before the child turned 19 years of age) and your child is wholly dependent on that person for support.

https://www.canada.ca/en/revenue-agency/campaigns/about-canada-revenue-agency-cra/common-law-married-separated-your-marital-status-changed-please-tell-canada-revenue-agency.html

I do not live with the father of my children.  Are we considered to be common law?

No you are not common law if you do not live with him. If you used to live with him, you would now be SEPARATED. Please see this Canada Revenue Agency (CRA) link to clarify your status.

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