The thing that is causing your swirls are bad towels. Invest in some microfiber towels and you wont get the swirling. I have two wash mitts from properautocare.com and they work great.

Now to get out the swirls I suggest starting with something as light as possible, such as just a simple paint cleaner (ie, megs step 1, in their 3 step system), before moving on to something coarser. See if that works first, because I am getting the impression that your swirls are light.

IMO, i'd stay away from Turtle Wax and Black Magic, these are not very good brands when it comes to product. Mother's and Meguiars are brands you should stick with.

Finally, waxing doesnt help your swirls, it simply fills them in and makes them harder to see, but when your wax wears off then the swirls will reappear. I think its better if you just try to get rid of them. The important thing here is usually quality application pads and buffing towels so you're not recreating the swirls as you're trying to removing them.

Okay that it is all.

EDIT: I'd also like to add that is basically impossible to burn your paint with a random orbit polisher.