Please give me advice 🥺need someone to talk to 😞

I’ve been with my child’s father for the last 3 years. We broke up the same week the baby was born. He hasn’t really supported me at all throughout the pregnancy or bought the baby anything, and he drinks literally every single day. He’s an alcoholic and kept promising he was going to quit for our child. He spent the first few nights over my house when baby was born and he was terrible at helping. I went downstairs to pump milk and told him to look after our baby and he just slept through the entire time the baby was crying. This went on for like 10 mins and I just thought he couldn’t calm baby down so I came back upstairs to help him only to find him completely asleep. I was pissed because I was already up the whole night doing everything myself. We had baby’s first doctor appointment that day and he said he was going home to assemble a crib with his brother and would be back to take us to baby’s doctor appt because i didn’t have clearance to drive. 20 minutes before the doctor appt he texts and saying he’s not coming back because he “ran out of time”. I’m calling him repeatedly and he doesn’t answer the phone. I had to drive the baby myself which was overwhelming being only a few days PP. I really feel like he used this as an excuse to get away from me and baby so he could stay home and sleep and drink to his hearts content. This is when we split up.

Moving forward he still hasn’t come to see our son or buy him anything. He came ONE time with his mom and the baby was sleeping the whole time. They stayed for an hour an left. He hasn’t tried to see his child since and then recently requested DNA testing in response to me filing for child support. This DNA test is completely shocking as he knows we spent all days together with no time apart. Honestly I saw his mom texting him back when I was pregnant and they were shit talking on me and she told him he should do a DNA test. I think it’s all bs but I’m willing to do it to prove them wrong. Anyway it’s been over a month since he’s seen the baby (with his mom) and baby is 2 months old now and he’s asking to come over and see the baby with his mother. Would you allow this? I feel as though since his mother didn’t support me through pregnancy and buy my baby anything and she wants DNA test and she says terrible things about me that I don’t feel like she should see my child right away for that reason and also the fact that the father hasn’t established a relationship with the baby. She also blocked me during my pregnancy on Facebook and said that she hated me and never wanted to see my face. I feel like he should come by himself to see his child without his mother and he should be lucky i even allow that as he is a deadbeat alcoholic. Am i wrong for feeling this way about his mom visiting right away??? What would you do??