I'm having flashbacks .......
Wow. It's been 18-plus years, but I can remember it like it was yesterday!
My daughter was a fussy one, too. She started crying at 12 days old and didn't stop until 7 and a half months. Like clockwork ... she cried every night for 2 hours.
OK, at 3 weeks postpartum ... you are TIRED. You are doing your best. And, you are doing it, mostly, alone. Yeah, that part I can relate to, also.
I'm sorry, I don't have any great advice for you. From what you say, I don't really think you're suffering from PPD ... more like exhaustion, frustration, lack of sleep ... all of that is normal. NOT FUN ... but it's normal.
You're right about the fact that your stress is communicating itself to your baby. (If you have a second child, you'll be amazed at how much easier it is) But that doesn't help you right now.
You need some relief. Is there someone (Grandma ... a friend ... anyone?) who could take over for a few hours, so that you could catch a shower and a nap? If not, then your baby's Dad really needs to step up ... job or no, it's his kid, too. He should be sharing in the work and loss of sleep, as well as the joy of his new child.
I'm sure this will ring hollow, but your baby WILL outgrow the fussiness. It may be several more months, but she WILL.
If you need to see your doctor, or you think you need counseling, go. I actually don't think you are anything other than a normal stressed-out, very tired new mom, but of course I could be wrong. And your baby sounds entirely normal and fine, too.
Not all babies are angels ... and not all moms have a perfect maternal instinct. Most often, it's a case of learning from your mistakes. And the firstborn gets ALL the mistakes! But, guess what ..... that baby usually turns out just fine ... whether because of us or in spite of us!
Good Luck ... Best Wishes ... been there, done that.