You missed the window. Any attempt you make to get her contact info that doesn't involve talking to her directly will be viewed by her as creepy. Just my opinion tho.
Don't sign anything until she gets a paternity test done. You might have to wait until it's born before you can stop worrying about it tho, which sucks.
Show genuine interest but stay cool and don't fall over yourself trying to please her. Do what you normally do, because that's probably what got you the date in the first place. Source: my hot party gf is spoiled as Fuck.