Check when you make your healthcare contribution to calculate your length of coverage. When I left my job in the US my healthcare didn't expire until 2.5 weeks later as I was paid twice a month in arrears and received a pay check, from which I made a healthcare contribution. This might be a solution?
Sorry I don't know anything about the current Cobra system or Obamacare.
I wouldn't quit a job and not have healthcare, my daughter caught meningitis when she was covered by Cobra after ageing out of my company plan, and before she was covered by her own post graduate employment. If she had not had coverage the bills would have been in the tens of thousands of dollars, as it was we paid $0 for her hospital admission, diagnosis and treatment. Cobra was expensive but in our case saved us huge bills.