Filing children as dependents

I am trying to get a level on the situation I have, currently the custodial parent has majority of time spent and taking care of children, and I see them when I am able. After going to tax offices getting understanding on what my situation is, I was told to the effect that because the custodial parent is on state disability and funded entirely by the states hand in support and makes no income outside of state support, that she cannot actually claim the kids as dependents, that it proves void unto me to claim the children, as well I am paying child support, we are separated on the way to official divorce. 

 

I question this a lot. This is Oregon state law, I am filing in Washington. Thank you for your responses and time.