how does DNA testing work?

If you do it while you're still in the hospital its free. If you've been discharged I think it's 200$ or so.

ok thank you. does it cost thru the state

Geez I can't even remember. With my first it was done through the state and I wanna say it was maybe a week or 2... wasn't very long.

how long do the results normally take

If it's after birth, it's just a cheek swab.