If you're having issues like your chest dropping or your arms not being in a strong stable position in your catch position in the snatch, it might not be an upper body mobility issue.
Widening your feet in your catch position can allow you to make room for your hips and allow you to keep your chest up. Resulting in a much stronger overhead position.
Try taking a plate and placing it between in your feet before you start warming up your snatch. Because of the plate, when you snatch you won't be able to keep your feet to close.
Once you're comfortable with that, remove the plate and keep practicing jumping your feet out into that better foot position.