If you set your images as exclusive, you cannot sell them through another agent. You can however sell these images from your own personal website and you can also post them on your Facebook profile. When it comes to social platforms, our recommendation is to read terms very carefully to make sure the images you post cannot be licensed, downloaded or sold from and by these platforms.
You can also shoot images based on a contract you have with a direct client (the newspaper in your case). If the newspaper is the end user - it is not a press agency for instance that may license/sell these images - then you can set the same files as exclusive with Dreamstime.
You can read more about the contributors' obligations regarding exclusive and non-exclusive images here:Terms here