I had an 19 month gap between my two daughters, my first had reflux. It was the hardest thing I've ever coped with. She got better when I started weaning at five months.
What reflux meds is she on? Has it been properly diagnosed?
Firstly you need to get some uninterrupted sleep, sleep deprivation is torture and aggravates any other conditions. Could you get your husband to take them both for one night and put some ear plugs in. I know it is hard to switch off but you need this. Or could you get someone to come round during the day so you can catch a few hours sleep?
I gave in to co-sleeping. I had a cot (with the side removed) on one side of the bed and a bedside crib on the opposite side. I also used formula so DH could feed the baby (obviously not everyone's choice but it meant he could do more). Oh and we all slept in sleeping bags, me DH and both kids. So no one got cold and there was no danger of a baby getting trapped under the duvet/blankets.
The house can wait, I know about laundry with a reflux baby. Can you concentrate on just getting that done and sod the housework.
I can't offer anything else but the reflux will pass, and then things will get better. For me it was the seven month mark when things improved, I know that seems an age away but you can do it.
In the mean time keep bothering the MH team, you need help. Is there a local childrens centre near you, talk to your HV sometimes they can get someone to give you a hand for a couple of hours just to get the pressure off.
You are under a lot of stress, keep talking. We're pretty nice over here x