Not sure if this is any help, but my DD was similar. Once she got out of that sleepy newborn phase, she could easily be awake most of the day apart from a few post-feed snoozes here and there.
I finally figured out that it wasn't that she wasn't tired, it was just that I wasn't helping her to fall asleep. I thought she'd just drop off when she was tired, so was just going through the day without actually encouraging her to nap.
I then read that a baby that age can't stay awake much more than 2 hours at a time, so started looking for her tired signs and encouraging her to sleep, which for my DD turned out to be every hour and a half. Once I started to encourage her to go to sleep (feeding, rocking, shushing etc), she went from sleeping about 2 hours max in the daytime to sleeping 4-5 hours, practically overnight, although the first few weeks were definitely harder as I was still learning her signs and she was still learning that sleeping in the daytime is a good thing.
She still can't stay awake for much more than an hour and a half at 3 1/2 months now.
As for waking up as soon as you put her down - my DD did this as well. I took the attitude that the first priority was to get her to sleep at all in the daytime, and then we could work how she was falling asleep and staying asleep. To begin with, she would only do 20 minutes, but is gradually getting longer, and will now easily do 40 minutes on her own, and sometimes even an hour!
I spent a couple of weeks noting down her sleeping times and the amount of sleep she got in a day, along with any major crying/fussy periods and her general mood. On the days where she got plenty of daytime sleep, she would be happy and content right up until bedtime, but the days she didn't were a recipe for the dreaded early evening fussiness and lots of screaming!
Good luck with it though - it's definitely worth persevering, even if it takes a few weeks!