I too own a new Mustang, a 2001 Laser Red(about 7 months old now). I also have had a few light swirls/scratches in the paint.
I am glad Ron K saw your post first, as I was about to suggest the same thing he did. Most dealers will remove the plastic transit coating from your vehicle, then wash it, and then detail using some kind of spray wax/carnuba based wax.
There may be two (or more) causes for your swirls
1. Improper washing techinques at the dealership, then the dealer applied a spray glaze or wax, this glaze filled in the swirls the dealer had caused. When you washed your car, you removed the filling properties of the wax, then you suddenly noticed them. IE they were there from the day you drove off the lot.
2. A poor quality terry cloth towel you used for drying or perhaps your wash mitt, may have marred the surface.
A couple of points to consider-
1. Examine your swirls after washing and drying the car, run your fingernail along them lightly, can you "feel" the scratch with your fingernail?
2. You may very well have a coat of light spray wax on your car. Without going into a lot of expense to strip the wax, try washing with a light solution of Dawn dishwashing liquid and water. After washing with Dawn, then rewash using a good quality carwash solution(Meguiar's Gold class, PS21, whatever...).
The Dawn will strip off the dealer wax, and then really let you look at your car's surface.
You will need to strip off the dealer wax anyway when you plan to "remove" the swirls. Why not see where the swirls are really located? Of course, once you remove the swirls(if needed) you will then need to rewax your car's surface again.
I would first get the dealer wax off the car, then look at the REAL surface. That can help to tell you if you have marrs in the wax coating, or acutal marrs/swirls in your clear.
I can not emphasize enough that the surface you are dealing with is VERY delicate. Swirl removal can take time. The folks on this board are very knowledgable, but DON'T try to tackle a problem until you are sure of what the real problem is. Trust me, I have done it in the past, and have often not been happy with my results.
For example,
No need to go attacking swirl marks in the clear(with a 3M product, or Meguiar's), if all you have is marrs in your wax. You might end up creating more scratches than you started with. Remove the wax first, then really see what is underneath. Then if you still have swirls, make a game plan.
I know this is long, but post back to me if you like. Once you know what you have, you can develop a better game plan, and ensure that you have a better shine!
SJ