You will feel much better very soon. C section stuff is pretty much sorted by 6 weeks so this problem is very short lived. By three weeks post surgery you should be capable of looking after a baby for 3.5 hours.
Sleep deprivation is a bitch, but tbh it will improve over the next few weeks as well. You should be able to catch some sleep during the day when baby sleeps. (And yes, I know, dd1 didn't sleep during the day, and ds1 fed every two hours day and night for ten months - I get it. But your wee baby is three weeks, and there is every chance that yours will develop a routine. And yes, I had a c section. I also had a third DC with brain damage who we had to feed at night because her muscle tone was too high to allow a swallow during the day, so we went nocturnal).
Time to woman up. And get your husband to man up. If you are bottle feeding, there is fuck all reason why he can't do half of the night feeds. You go to bed at 8pm and hand the baby to him. He deals until midnight. After midnight, you get to deal. Or vice versa, but at least you get an unbroken 4 hours or so.
And don't take sleeping tablets with a newborn, that's not a great idea.