Lots of babies do this. Sleeping together is often the best way to maximise the amount of rest you get - it's still tough, though. Getting up and down all night means even less sleep. Could your partner sleep elsewhere for a while? That's what we did.
Food doesn't help. Both my kids did this for quite a while. You survive it and it does get better, but it's hell while you're in it. What helped me was to let go, stop trying to fix the unfixable (nothing wrong with the kids, the problem was mine) and just do whatever helped me. So bed sharing, sleep ins on weekends when my partner was around, rest whenever I could, and so on.
Child care is accomplishing something! Don't write yourself off