A week is not much time to acclimatize. We adopted two sisters who, when visiting their foster home, did not show their face to us. At all. The foster mom said that they do come out for food, but they will wait it out if any strangers are present. We adopted them anyway because our mind was already made up.
We kept them in one room for two weeks, visitng every day. We even got to give them treats after the first week, but whenever either of us moved (even wiggled a foot) they'd back off. The only movement that didn't startle them was playing with toys or giving treats.
After two weeks we opened the staircase to them. It took them over a day to even want to explore that room. When we eventually opened the whole house to them, they weren't particularly interested in going downstairs. They would only go there when we were down, but they wouldn't go together with us.
All in all, it took them two months to want to walk around in the house, and maybe six weeks more before they actually used the downstairs as a living space rather than a rarely visited "outside" area.
Of course, your cat may be much quicker. Ours still shy away from visitors (or cars coming near the driveway) and that's after two years now; so I'm aware our cats are more shy than the average cat.
Since you didn't confine her to a room, you've given her ample room to not deal with you if she doesn't want to. That's okay, but it means it might take a bit longer before she has enough experience in dealing with you that she starts trusting you more.
She uses the litter box, she eats, she explores when unsupervised. You have a healthy, functional cat. Just give her time (and maybe lure her with treats) so she can figure out that you're not a competitor but a friend. Try to avoid instigating interactions, as this can cause them to back off and trust you less. Not so much because you risk permanently breaking something (you won't unless you take it to extremes or you have a pathologically terrified cat), but rather because regression means it will take longer overall for her to get used to you.
Based on already having pushed the topic with her in the week she's there, I think you're either being impatient or have misjudged the timeframe it takes for them to adjust. I would give her a month, up to three if you have a relatively shy/anxious cat, before I'd worry about them not opening up.
It's only been a week, but she's a darling and I'm eager to pet her. Is there anything I can do to make her more comfortable, or anything I shouldn't do?
The best way to safely approach a cat in this situation is to let them make the first move. This means you ensure that they consent to the interaction and thus won't blame you for approaching them.
The simplest way is by offering treats. This gives the cat a choice:
- Stay in my comfort zone and have no treats.
- Approach the human and get treats.
Even if she doesn't go for it initially, you're creating a rapport. The cat observed you and noticed that you weren't a threat or too imposing for her. The longer this goes on, the more that the cat will subconsciously forget why it's choosing to not come closer for the treat.
Essentially, they will eventually think "they haven't harmed me at any point. Might as well go and get the treat".
When she does, keep this schedule up. You're not there yet. You've broken the first layer of ice. Now, what you do is repeat the ritual to reinforce the "Marie = giver of nice things" habit.
Eventually, you can start using the treats to get them to do what you want. For example, I gave them half of their treat (3 pieces of sausage). The other three pieces, they only got after petting them once.
This repeats the same pattern: they were initially unwilling, but eventually realized that "getting a pet isn't so bad and hey, we get some treats!". They took it far enough to come to me to offer pets (pushing themselves into me) because they assumed that would also give them treats. And for a little while, I did (because I wanted to reinforce it).
Once you feel like you've sufficiently reinforced the behavior that they needed to learn, you can move on to another step, e.g. putting a treat on your lap so they come closer. But whenever they hesitate, make sure that the old deal is still valid, so they don't feel like they've lost the previous deal.