The good news is that you can clone those spots out so your photos aren't completely useless.
If it were dust, it would be a consistent shape and size. The only thing that would change is how well defined it is based on lens aperture. I may or may not move around the sensor, depending on what you do between exposures. It most definitely would not show up white however.
Yes, I think it's ridiculous to have to pay for shipping but it's unlikely you're going to get around it. I still think you should try for an exchange from the place of purchase and let them deal with it. It isn't as if you've changed your mind about the camera, there is clearly something wrong with it and you've had it less than a month.