Your sentence is mostly clear and correct! However, here is a slightly improved version for better grammar and clarity:
Original sentence: Puffins make nests digging shallow burrows. Sometimes, they move in empty homes made by other birds and animals, such as petrels or rabbits.
Corrected sentence: Puffins make nests by digging shallow burrows. Sometimes, they move into empty homes made by other birds and animals, such as petrels or rabbits.
Explanation:
- Adding by before digging shows how puffins make their nests.
- Changing move in to move into is better grammar for entering a place.