If you were still married (not legally separated) as of 12/31, you have to file 'Married Filing Jointly' or 'Married Filing Separately'.   Filing Jointly is almost always the better option.  You can claim the kids as long as you meet the requirements.   It depends on their age, earnings, and a few other factors; their Medicaid status does not matter.  

Who can you claim as a dependent?  https://ttlc.intuit.com/replies/5187060