L.S.
I can soooo relate to what you're going through. My son didn't have reflux but was allergic to his crib. My advice to you is to get a sling such as a Maya wrap, lucky sling, etc.... These are much better than the baby bjorn for hit hips; however, that would do in a pinch too. You can put the baby in a position so he is sitting up and let him fall asleep if need be. This way, you can walk around and get things done.
The pediatrition in the hospital suggested this to me when my son was born. She said that some babies are just like this, they need to be held. Let me also say that this is OK, you are NOT spoiling your child if you do this. I should also note that my son will be 2 in a few weeks and he is very, very independent and not at all clingy so the idea that picking a child up or holding him all the time will make him clingy is not true.
The other thing I would do was to move my glider near the tv or a good book and just let him nap on the boppy and me. This was hard to do at first having been a workaholic, but I soon grew to enjoy it. I got to catch up on a lot of reruns, good books, phone calls, etc.. This may not work for you, but I thought I would just float it out there. :)
Enjoy the time that you have where you can hold him. Beleive me, it will be over before you know it. You'll always be able to catch up on housework later, but you will only have this one time while he's young.