This sounds just like my kid - though since I had only the one I thought everyone went through the same stress. It does get better!
(There's a light at the end of the tunnel - if they'd turn the darn thing off maybe you could get some sleep.)
Anyway, here's what I can think of that may help.
First. If you really want to breastfeed, just get rid of the formula now. The formula may be aggravating his symptoms, and it will certainly undercut your production of milk as well.
If you are concerned about the quantity of milk you have, that can be improved quickly.
To begin with forget everything they told you about switching the baby around while you are nursing. Your breastmilk changes in kind as the baby nurses. The first milk is thin and watery, suitable for keeping a baby hydrated, but not really food. The more food-like milk kicks in after the baby hs been nursing for a while, and the last of it is really thick and creamy. This hind milk is the good stuff, and if you keep bouncing him around he may not even be getting to it and just filling up on the more watery foremilk of both breasts instead. In that case while he seems to be nursing a lot, he is never getting to the richer, more satisfying hindmilk. Also his nursing is never fully emptying either side, and therefore not signaling your body to make more milk.
Anyway, let him nurse on the first side until it runs dry. If he is still hungry he'll start fussing when he's not getting anything. At that point switch him. If he doesn't completely empty the second side, you need to pump it completely dry. This will help prevent infections, keep you comfortable, and increase your milk production. (Supply/demand can be your friend!) Babies have sudden growth spurts where they are ravenously hungry. If you are regularly expressing extra milk, you build up your supply in advance, and can absorb these little bumps in appetite easily.
Another benefit is that you can bank the extra milk in the freezer so that your husband can use it in a bottle when he spells you in the evenings.
Second. As above, your husband should be able to spell you through an entire feeding/sleep cycle. Pick your time - post dinner to midnight? - and just leave him to it and go sleep. He will have bottles, and he can handle things. You can put in ear plugs if you have to, but get an uninterrupted hunk of sleep.
This will knock your sleep cycle off kilter for a while, but the baby's extreme unhappiness won't last forever. You are just trying to get over this rough patch. Later, when things smooth out, you can take turns doing a night feeding as needed, and get everyone back to a normal bedtime.
Right now, your body is telling you it wants sleep, and your husband has a work schedule, so you are just trying to make it through the best you can.
3. Just a thought. With my kid I wondered whether the amount of milk I was drinking m was effecting my breastmilk. (It's a liquid, it's a food, it take no prep time - what's not to like?) Since I too was at my wit's end after a few weeks of this, I looked at everything I could find trying to change things. I read that milk proteins could be turning up in my breastmilk and causing the colic, so I stopped drinking milk. After three or four days I didn't notice a difference, so I gave it up. Years later I read that it would have taken weeks, not days, for my system to clear out enough to see any change. To this day my kid just doesn't like milk, though she doesn't have any big allergy symptoms. I continue to wonder if it would have helped if I had stuck to it.
Anyway, you might research and see if there is anything in your diet that could be triggering a reaction from your child.
It will get better.