When does it get better?

Writing anonymously because I'm ashamed to feel this way. I need advice, reassurance, anything.

I am having an almost impossibly difficult time with my 7-week-old son. He won't ever sleep longer than 2 hours at a stretch. He is grumpy and fussy all day (which makes sense, because he's exhausted). He absolutely will not let me put him down - he cries hysterically when he's not being held.

I'm sleep-deprived beyond the healthy limit. I'm frustrated. I'm fighting with my husband almost daily. Sex is completely out of the question because of fatigue (and because there's a baby attached to me at all hours), so we're distant. I'm lonely because I can't see my friends, because no one wants to hang out with a screaming baby. I can't find time to eat or shower (much less do hobbies I enjoy) whenever my husband's at work because the baby just screams when I put him down. I'm terrified to go back to work in a couple of weeks. I love my son very much. But I feel like I'm at the breaking point.

Is this normal for a 7-week-old? Is it just expected of infancy and his temperament? If so, when does it get better? I feel like I'm not enjoying this time with my son, which breaks my heart for me and for him.

(This is not depression. I've been through depression before. This is fatigue and frustration and feeling completely overwhelmed.)