You have received some good advice here, OP. However, not all of it chimes with my experience as a new mum.
My 9 week old baby doesn't especially like being put down either. I never leave her to cry for longer than a minute or so, not because I think it will harm her but because in my experience she doesn't calm down or stop, so the only thing that is going to work is giving her what she needs. That involves lots of cuddling, rocking, holding etc. But it doesn't always. Sometimes she wants to be put down and that is why she is crying! Sometimes she wants me to give her some attention and sometimes she is overtired and wants a reduction in stimulation, so dimmed lights, TV off or turned right down, toys away etc. The only thing that has worked for me is to carry on trying putting her down - sometimes it improves the situation, sometimes not.
I have to say I haven't found some of the 'cuddle, cuddle, cuddle - enjoy it!' MN advice that helpful
If I did that, I a) wouldn't enjoy it, because there are times my arms are so tired I want to cry, and I don't find that enjoyable and b) the baby wouldn't be in any sort of routine at all.
And she is in a routine! Even at 4 weeks I had some elements of a routine in place: wake up, change baby, feed baby, settle baby for a short nap if possible, clean up detritus of the previous night (nappies etc) and wash and dress baby if not sleeping, eat something myself, wash, feed (and usually change) baby, try to get out of the house for an hour or two (cafe or book shop) or watch some TV, eat lunch, do some housework, feed and change etc.
And all of that took place amidst much rocking, singing and comforting of the baby! But if I had had aimed only to brush my teeth in a whole day and just constantly 'cuddle, cuddle, repeat' I can honestly say I think I would have gone nuts.
That isn't advice. Just my experience.