Cant help you with the snatch because mine is terrible. But with the jerk I prefer to split. I can get lower and feel a lot more stable than when I push jerk. If you feel unstable in the split Id bet you arent splitting wide enough and you are letting your knee get ahead of your front foot.
The way I worked on these issues was to take a pvc/broomstick and lay it on the ground in between my feet and then just split jerk with an empty bar over and over again. Just focus on keeping your front shin vertical and also keeping your feet wide enough that you arent feeling any side to side instability. When in the split you should feel firmly planted in every direction.