Unmarried parents custody. Thanks for answers!

So I'm the primary caregiver for my child, he sees his dad 1 maybe 2 days a week at most for a few hrs. Me taking him to his house and picking him up if I don't stay during the visit. That's at his will even though he lives 5 mins down the road. We've never been married, but father is on the birth certificate. Do I automatically have full custody since we've never been married? I was thinking about putting him on child support and was curious about this. I think he is unfit as a parent and don't trust to leave my child with him especially for long periods of time. Mainly at nap time is the only time he has me drop our child off, or if he's going to his mothers.