Can I get full custody of an unborn child and how soon after she is born?

Ex-girlfriend is pregnant with my child. She has two other children all from different men. The other children were taken from her and their fathers have full custody of them. She doesn't even have visitation rights. She lives with her father that is a registered sex offender for molesting my ex from ages 7 to 9 and he was in prison for about 8 years. She has never been through any counseling to deal with her issues. She has a history of moving from man to man and most have had drug issues. She has been arrested more than once for drug related issues. She has also had an abortion in the past. I feel she is too unstable and she has no place to live except with her father or someone she is dating. I do not want my child with a man she isn’t married to. We are in the state of Louisiana. The men of her other children have advised I get full custody of this baby. Also she has told me that when she was a child she took her guinea pigs and drowned them in the bath tub then placed them back in the cage so no one would know. That seems very disturbing to me.