Probably not much point trying to put her down once you've got her to sleep if she wakes after 10 minutes. Ideally you want them to fall asleep where they're going to stay asleep.
I'd probably try to get the overtiredness sorted and into a bit of a rhythm for naps before working on getting her to nap in the cot.
You could do something like
awake at 7am and feed
8am into the sling with a dummy, white noise and pat her bottom as she falls asleep
when she wakes from that one eg 9am feed and get both of you changed and dressed.
Into the pram with dummy/comforter/white noise and out for a walk by 10.30am
Even if it takes 20+ minutes for her to fall asleep I'd persevere with it and then once asleep keep walking for at least 45 minutes.
Then you can hopefully head home and either bring the pram in or leave it by the back door. Sit down and relax and hopefully she will sleep another hour, but anytime she starts to stir just rock the pram again!
Pram naps are great at this age as once they are used to sleeping there you don't have to go out for a long walk, you can just rock to sleep in the hallway or garden.
Once you've hopefully got one long nap in it's fine for the afternoon naps to be shorter sling naps. Or if you have a bouncer chair or swing you can try using them for naps too.
Only once you've got her napping regularly and hopefully having say 4 naps a day using all the comforters and patting would I try having naps in the cot.
Better to have plenty of good naps in the pram, sling, bouncer etc than worry too much about the cot.
At that stage you could then try taking her to her room at the established nap time, putting the white noise on and rocking/bottom patting til drowsy then putting her down and continuing to shush and bottom pat til asleep.