I'm currently a Jamaican immigrant living in the US. I'm not a green card holder nor am I a citizen but I do currently have DACA. I have family living in Sweden and would like to live there permanently. Would I need to obtain a visa from my home country and have it shipped to the US in order to travel to Sweden? Will I have any complications such as deportation at the airport?
You need to work with the embassy of the target country you want to move to, in the country you currently reside.
So in your case, that would be the Swedish Embassy in the US.
Traveling back to Jamaica would not help you.