I was hoping to hear back from you with additional information, but I also didn't want your question to go unanswered.
It sounds like she's doing very well, coming when you call and sleeping with you, considering you only have her for 3 months. Her innate personality sounds skittish and shy and she's not comfortable with anyone else in the house, but you. Most cats hate visitors and will 'go for cover', so don't think this is odd.
I'm so sorry you got scratched after she heard a noise that startled her and you were holding her. I hope the scratch wasn't too bad and you applied first aid.
I would recommend not picking her up, so this incident cannot recur, for the time being, anyway, and also, get a Feliway plug in diffuser for a couple of rooms in your house. This resembles a plug in air freshener, but has no scent to humans. The diffuser emits calming cat pheromones that help ease stress and anxiety.
Make sure she's not bored and get a tall cat tree or kitty condo, if you don't already have one, get some scratching pads for the floor and also, get some great toys to keep her occupied. The list below is my favorite and you can check them out online and/or in your pet supply store. As you can see, there's also a cat dvd, that she'll love. I feel if her environment is enriched, she will feel calmer.
This is a process and will take some time; she's still adapting to her new home and she's going to have to learn that every noise and every person does not mean 'danger.' It will happen, but you have to give her some time and continue to be loving to her, pet her, speak to her gently, repeat her name a lot, that will make her feel secure. If she feels running under something when she hears a noise or a visitor, makes her feel safe, this is instinctive. She'll soon realize that she doesn't have to do that, if no harm will come to her.
Make sure her claws are well-trimmed and if you can't do it comfortably yourself, you can ask your vet or a groomer to do it. If she has a scratching post (the kitty condos are covered with carpet and double as a scratching post) or the scratching pads I mentioned, which go on the floor, she'll be sloughing off her old nail sheaths and keeping her claws in good shape. The very point will still need to be trimmed every 2-3 weeks, though.
Start offering her cat-healthy treats from your palm--not too many, though, and then, when you have guests, if she sticks around, they can take turns offering a treat from their palm, too; however, if you feel she's not ready for this, wait; you don't want her to scratch any of your guests. The way to a cat's heart is through her stomach! : )
I hope all will be well with your furry girl!
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