Lack of sleep is killing me

So my daughter is 6 weeks old (born at 35w). She is EBF and feeds very often during the day and night. She is growing perfectly and her weight gain has surprised the doctors and nurses at her appointments.

Here’s the thing - she has some reflux (we think caused by a milk allergy, I have been told today to try and cut out dairy in my diet. ) and after each feed we have to sit her up for 30 mins. However, even after we put her down to sleep, she will immediately wake up and squirm, then will vomit or burp. Because she’s awake from this, she will want to feed again or be really fussy and the cycle continues. I have tried raising her crib head (advised by medical professionals), and so many other things just to try and help her, but I’m absolutely suffering with the lack of sleep as I just can’t put her down at night long enough for me to get some sleep. During the day she will only sleep in our arms or in her baby seat thing. My husband helps out as much as he can but he works a manual job and needs more than a couple hours sleep at night. We have a 4yo who I feel like I’m constantly shouting at due to being irritated and sleep deprived and I’m always falling asleep with my baby in my arms which I know is so dangerous, but I’m genuinely so exhausted that i fall asleep as soon as I’m in a comfy position. I’m losing my mind, am feeling so teary and agitated and just need some advice or words of encouragement that things will get better.

Edit - thank you for all your helpful comments 😊 will see what cutting out dairy / reflux medication will do! x