I haven't gone out in over 3 years.

This is so embarrassing. Problem is how title reads. I work a full time job, and I'm a single parent of a 3 year old daughter. Her father was crazy and mildly abusive so I left him when I was 7 months pregnant. He didn't sign her birth certificate and has continually told me we aren't together and has dated multiple other women. I haven't been able to take me time or a day/ night to myself because of him. He told me if I date any other guy or go out at night or day he will sue me for custody and tell the judge I drop my 3 year old off at my parents to have sex with other men. He has always told everyone I have sex with other men in cars and I don't take care of our daughter. This is not true and I have lived in fear that if i take one day or night to myself the judge will take his side. I work 3rd shift at my job and I work hard. I work about 60 hours plus a week to support me and my child. We get no public or state assistance. My mom stays at my apartment while I'm at work and watches my daughter sleep until I get home. I am always very timely getting home. I keep a record of everything because I'm so scared of these accusations. I only go out to grocery shop, or take my child somewhere. I haven't take a day off for 3 years. I make sure I'm always with someone ie; my parents or daughter who can verify where I am at. I can't date because if I date a guy and my ex finds out He will say I have sex with the guy and abandon my child. This fear is eating me away. My friend had a bad custody battle and said her ex husband accused her of things like this and was asked why couldn't the husband have the child if she was our having sex when her ex husband had no proof she was doing it! What do I do? I suffer from depression and obviously anxiety but I'm sick of living in fear of this.