Please help. Baby tossing and turning all night.

We recently moved our baby to the crib and although she is not crying she’s tossing and turning. There’s a period of time where she’s perfectly fine and then she will start making noises and moving around a lot. Tossing her head and body from side to side. Is she uncomfortable? I’m almost tempted to take her out 😭 also I did notice her face and hands are cold but her neck and back is fine. She’s in a long onesie pajama and a light sleep sack...

I know how important sleep is for babies and I’m afraid that her tossing and turning so much is not allowing her to get her best sleep.