There is no one perfect way to do It, it depends on what you are trying to achieve.
Do you want money, if so take it to the butcher. Maybe it is a 3 star and you need the pelt then skin it and sell the carcass. When skinning you also get meat and normally another part of the animal, either keep for crafting /cooking or sell for money/donate.
When taking stuff to pearson/camp there are 3 options. Crafting,provisions and funds, when donating it will tell you what it will be used for when selecting the object you want to donate. So if you have a perfect pelt pearson can use for crafting it will say it is going to be used for crafting. You can bank perfect pelts/crafting items with pearson, so once you have donated all the crafting items pearson needs any future donations will be for funds.
To skin or not to skin is your choice. If i am going to just sell or donate a animal that i put on the horse i don't skin it. if i want to donate the meat but sell the skin i will skin it. Also i have nothing to back this up or done any testing but i think it takes longer for the animal to spoil when it is not skinned, just my opinion.