Need legal advice to know what I can do about full custody (kind of long but please..help)?

I don't like putting my business out there but I need to get advice before I can really go do anything..

Okay, well I am a single mother to my 15month old daughter. When I was about 7 months pregnant my baby's father literally walked out on us and we ended it completely when he tried to come back. The day before I went into labor he called me talking a lot of b.s then gave me his rights over the phone with two witnesses. Also, the day I was leaving the hospital, he called and did the same thing. After that he wanted to see her. I told him I wanted him in her life so I didn't deny him the right. He waited til she was 3weeks to see her. After that, he never came again. While I was pregnant to now, he never paid a dime for her. I paid for the doctor visits and my mother paid for the hospital. And he took 400 I saved for my daughter. And from the third month I was pregnant, there was all types of abuse going on. He also threatened me and my boyfriend that has been there for my daughter since she was born. I have not had contact from him in a year or so but I do worry of him deciding one day to pop up. I personally don't want him to have the right to pop in and out her life while I have been a single teenage mother working hard to be there for her one hundred percent but I also don't want the possibility of having joint custody. If he could abuse me while I was pregnant I can't trust him with her. What should I do? Try to serve him and get full custody? Leave it alone? Do I have the chance to get full custody or is there a big possibility of joint custody? I need these questions answered please...

Thank you. I know this was long but I appreciate it all.