Baby daddy drama

Need advice - me and the baby’s dad have not been together during the whole pregnancy. He denied the baby until proven with a dna test. He’s not offered to pay for a single thing for the baby, and yet has all these demands etc the baby has overnight stays with him straight away. His actions and words don’t meet up, and every time I feel like we have a positive conversation something comes out the woodworks and sets us back. For example he works away, and I’ve been accommodating and updating him with everything he needs to know. We had a phone call as he couldn’t meet in person to discuss things due to work, and I found out later that night he was actually back home in the area, and was the following weekend aswell. It’s mentally draining me being lied to, and would rather he’s just upfront and honest. I don’t feel like he’s interested, and is only keeping contact for his family’s sake. I don’t know what to do, but the baby is due soon and It’s causing a lot of anxiety as I don’t trust him. His family haven’t been very accommodating or supportive but are expecting full contact and overnight stays with the baby after he is born. Does anyone have any advice for difficult co parenting?