Dear M.,
I know how exhausting/frustrating this situation can be! My experience teaches me that whatever works for your family is the right thing to do, so don't worry that you're doing something wrong.
When your son wakes in the night is he able to express what he needs or why he wakes up? Is there something waking him? Sometimes doing a very long or involved bed-time ritual every night helps with getting the body and the mind ready for sleeping, so...bath, toothbrushing, reading, cuddling, singing the same song or lullaby, etc. But make it a ritual that is the very same every night.
Is your 7 week old sleeping with you at night? If so, maybe your son is feeling left out and needs some extra reassurances or something special for him at bedtime so he feels he's getting some of that nurturing attention for himself too. Depending on how mature he is for his age, you may be able to make a deal with him that you will lie down in his bed while he falls asleep, and then he will stay there until morning. You might even have some little items that he really likes (stickers, little toys, etc.) that can reward him each morning that he stays in his bed all night. Maybe put them in a basket that he can pick something out of every morning that he stayed in bed all night. So it's a positive reward that he'll want to work toward.
Unfortunately some children do have a difficult time sleeping through the night for various reasons. I have an eleven year old who still wakes in the night 1-3 times per week. It's not at all uncommon, but in your son's case there may be some things you can do to help him develop habits so he can self-soothe in the night and fall back to sleep by himself. A special stuffed animal, blanket or toy? You can even try granting his favorite stuffed animal with magic "powers" that will help him fall back asleep if he wakes in the night.
Sometimes the noise of a radio or tv can actually wake kids up in the night.
I am a psychologist, not a physician, but there may be some homeopathic remedies or even warm milk before bed that would help him sleep through the night.
Even though you are very tired, keep in mind that most of children's behaviors are trying to tell us something, and our job as parents is to decode the message. Also, they grow so fast that most of these phases are temporary.
Keep your positive approach, and I wish you the best!
J.