You need time to catch up. What do you have planned this weekend?
You need to plan a nice long nap! Not saying this is the solution, but even a bit of sleep will help you think clearer.
What can you change to get more sleep?
Can you go to bed earlier?
How about you and the baby sleep in a different room, just for now.
Then DH can be rested, and have no excuse not to get up with the baby in the mornings and let you get a bit more sleep.
What things in your daily routine are getting in the way of you sleeping, apart from the baby? Who's cooking dinner for example.
Remember this isn't forever, you can make drastic changes now and change back when your baby sleeps more.
e.g. your DH could grab himself food on the way home, you could eat before he gets back then he could just take the baby as soon as he gets in, so you can go sleep for a bit. Maybe not every day if you value eating together, but say 3 days a week while it's so tough?
Can your DH take a day off, to give you a chance to catch up on sleep?
This need to be taken seriously, you need a chance to catch up. What other ways could you get a chance to grab more sleep?