Correct the whole point of conditioner is to remoisturize ur hair of oils after u stripped off everything with shampoo. I dont recommend putting it directly on your scalp its mostly the top layer that needs it. After u put a layer on just briefly rinse and the hop out. Also try not to use any sort of heat dry, if u can resort to the cool/cold setting of your hair dryer
You actually dont want to get conditioner fully out cuz then that would defeat the purpose of conditoner. It sounds like your scalp is overproducing oil to make up for the dryness. Try to slowly shampoo less and after a shampoo day use some kind of oil i like to use argan oil and on other days only use conditioner
If ur talking ab shampoo use whatever is best suited for your hair type just make sure there arent any harsh sulfates and parabens. I personally use shea moisture, their brand has a lot of options to choose from and they all good ingredients. I recommend getting the yuka app you can scan any of ur products and it will rate it. If ur talking abt body soap just whatever that doesnt have any harsh cleaning agents. Use the yuka app