This seems to be a stage that 4 year olds go through. My highly independent 4 year old has also recently started doing this.
Saying she is scared is to get attention, in one respect. But there is also the fact that they are going through some pretty intense psychological development right now, and is uncertain about the changes. They need that extra bit of reassurance, but they also needs rules and boundries. Try to find additional ways to help her feel comfortable.
In addition, she has had/will have two MAJOR changes to deal with at the same time as her internal development. She was just pulled away from all the comfort/routine/familiarity of her old home and expected to transfer it to a new place. Kids this age are big on keeping things the same. This is how their sense of trust is built. She will also have to get used to not seeing you much during the week. Expect some major melt downs around that.
A few things that help my daughter are:
1. After we snuggle up and read a book, we talk about what she did that day - starting with breakfast ("you woke up, ate eggs and toast for breakfast, then went to preschool, tell me what you did at preschool, then we came home, ate lunch, looked for lady bugs in the back yard, played with trains and puzzles, cooked dinner together, played with daddy, took a bath, brushed teeth, read a book, talked about what we did today, then went to sleep.")
More than just playing, you are focusing on her and having a conversation with her about what she did and treating it with as much importance as another adult (oooooo). The "then went to sleep" part is also a bit of a subconscious 'nudge' in the right direction too.
2. Snuggle her down in bed, then say "Let's think of five fun things you can dream about." This sets a positive tone for bed and makes her happy just to think about these things, plus you are again giving that subconscious 'nudge' when you talk about dreaming.
3. Snuggle her down in bed, then kiss all ten fingers and each palm. Tell her she can hold on to all those kisses for the night, and that she can have more kisses when she wakes up in the morning. (More nudging.)