In Arizona, Common Law Marriage is not recognized. I heard if a couple has a common law marriage and has kids, if the man dies without a will, the kids/wife don't end up inheriting his possessions. If both the parents died, the kids would be left to the state to take care of. Is this true? I did not find this situation explained in my Google searches.
Nolo appears to have a good explanation.
The probate court will have to confirm that the children are yours before they can inherit. Your children would inherit any separately held assets that don't have a beneficiary associated with them.
You should make sure your spouse (or kids) are the beneficiaries on any retirement accounts, bank accounts, life insurance, etc. Those won't go into probate and will be inherited directly. If your minor children are the beneficiaries, then the court will have to appoint someone to manage the money for them. That will likely be their mother/other parent, but the court would have to appoint that person.
For jointly held assets, that don't have a beneficiary, your common law spouse would inherit half the asset, your children would inherit the other half.
Property depends on the deed whether it's held as joint tenants or tenants in common.