Animating paths in After Effects
Probably the most dynamic and full control option would be to use taper stroke in after effects - just redraw the path and apply taper like here - https://i.ytimg.com/vi/jFBKMRncG80/maxresdefault.jpg
then you have a multitude of options to displace them (like wave warp, mesh warp, wiggle-shear, or even map the end points to nulls and use wiggle position).
Tapered stroke is only available in CC 2020 and 2021 if I remember correct.
Another option would be to expand shapes in AI and copy path into AE which you can then animate later, but it would be more time consuming.