You've had a tough time, and you are doing well to keep everything going.
How old is your baby? Do you have a health visitor? Just wondering how old your baby is, and if he might have colic, or whether health visitor or GP might be able to help some way. Some babies do just cry a lot, some turn out to have a reason. My eldest threw up all day everyday, woke up every hour or two at night, turned out to be waking up with heartburn as he had a reflux problem not properly diagnosed until he was about 9 months old.
Also wondering whether you have a friend you would trust to look after the baby even for an hour or two, to give you a break. It's hard enough work looking after a baby who cries a lot, let alone while you are trying to study as well, and you need a break sometimes. Even people with quiet babies who sleep to schedule need some adult company away from their babies from time to time.
Is there a Uni welfare fund that might help with childcare costs? Maybe your Uni has a creche or nursery that you could use?