Dropper seatpost oscillating - relation to rear shock
So you are basically saying the upper part of the dropper post is loose in the lower. 

There is definitely something broken, possibly a bushing between the upper and lower has broken or worn out. Without knowing the specific model it's not possible to give more advice, you may be able to find a manual for your specific model that provides a drawing or list of parts

The suspension will not be affected at all.