It's totally natural for her want to be close to you - she spent 9 mos constantly attached and she's only been out of the womb for 3.5 weeks. She's also in a big, scary and cold world. It totally agree that 3.5 weeks (much less even 3.5 mos) is way too young to let her "cry it out".
Have you tried a sling? That worked wonderfully for my second son and it leaves your hands free. For the oldest, I had one of those front carriers that didn't work too well, but he was happy in the swing. Son number two was like your daughter and nothing could replace his mom, so he spent alot of time in the sling. Once he became mobile (and bigger) we migrated to a backpack that I would use even around the house so I could get stuff done. He's 11 and is still very attached to mom, which I don't consider a bad thing at all - he's an incredibly sweet, loving, child.
That all said, it is okay for you to put her down for the 5-10 minutes it would take you to do minimal self care everyday. Just make sure she can see you and talk to her while you're doing your activities. As time goes by, you should be able to have more and more time to yourself.
It could also be that she's reacting to her diet and it's just comforting to be held. If you're bottlefeeding, she could be allergic to dairy or soy. If you're breastfeeding, she could be reacting to your diet. It turns out that both my kids (and myself) are gluten intolerant and it is quite likely that the gluten (and dairy with my first) caused alot of stomach pain in my kids. Also, my youngest son was very fussy if I ate broccoli or drank orange juice.
I know you have lots that you want to do, but I can tell you that they grow up so fast - it truly just seems like yesterday that my oldest was born. Enjoy these early precious days!