I finally went online and signed up for the part-timer's medical insurance at my work, and it is a good thing. You are required to sign up within 45 days of your hire date or else wait until open enrollment, and I would have missed the deadline if I hadn't started asking questions and not getting answers when I did.
When I got to work the next day I did fill in the new manager about this, and he made sure that packets were printed out and given to all the other part-timers that very day. I am the second newest employee though, so everyone else will have to wait until open enrollment if they want the insurance. That is only till May for this plan, but it would have been too long for me. My COBRA benefits run out only a few days after my new plan takes effect.
So my assertiveness (a nice way to say pushy) paid off in this instance.