Daisiemoo sorry you are going through this. My adopted ds is 4 (came to us at 3), has been home a year and experiences some of what you are describing.
Please do try and access some post adoption support, if you can. I know it seems early but you may have to wait for the support and best to get your request in soon, it is not a sign of failure to ask for it.
You are just over three months in and that is quite early for play dates, not sure how long we waited, probably about three or four months too. So I am not saying it is wrong, but if it doesn't work out feel free to stop them for a bit, or go to neutral places with fixed 'facilities' like a park.
So agree with gabsdot45 try not to overstimulate, limit toys, I know it is hard, our house is like a bloody toy warehouse!
Agree with SponsoredByTheBadFairy (excellent advice) model sharing. We can't really 'make' our kids share, we can show them how and they can learn or we can force them to give up toys, which isn't really sharing. I went through this with my birth dd who was an 'only' child until 9. She was really good at sharing which is why I get so cross when people say 'only children can't share!
Hypervigilance. Well not had exact experience of this but ds came with a word, the word was 'can't'! He could not do a lot of things and was clearly fearful of water, being cold, getting wet, hights (as in climbing equipment) etc. I devised a little song I could sing, "My name Dave, I am brave, sometimes I feel worried, and I won't be hurried!' Along those lines (his name is not Dave!). I would sing it softly to him when he got nervous and sometimes he would say 'sing the song!' I no longer need to sing it (thankfully) and he gets stuff done much more easily. He climbed on a walking 'don't touch the ground' trail today, I went mad with congratulations because he managed half of it and initially it looked too hard for him. But he did it 9or at least half of it!). So when she conquerors a fear, even a small one, lots of praise, but if she does not, no recriminations (I know you know!).