Substantial pay increase and insurance

My partner and I are not married. We have 2 children. We used to each claim one child when neither of us had private insurance through our jobs (medicaid.) My partner has been claiming the kids for the last 2 years to get the payment assistance for the insurance premium because he got a job that offers medical and we wanted the kids covered. I continued to be on state insurance until I landed a job in July that increased my income 150 percent. Since my partner has been claiming the kids, I filed my w4 as zero and now they take almost 40 percent of my income each check. I currently don't have insurance but will hopefully after the first of the year. What do you suggest for tax time ( i think if i try to claim the kids well have to pay a penalty for the insurance and if i dont ill contibur to hemmorage money) and what to change next year so we can keep as much money as we can through the year?