First things first, fisher. What's the condition of your paint? and is it single-stage or clearcoated? (P.S. Lots of us put our car year/make/model/color in our sig lines so folks will know those basics when replying to our queries.)
Deanski, I believe a PC is a "random orbital."
I've not just used 3M Fine Cut, but also their HeavyCut. I'm in the process of using the HeavyCut with a yellow cutting pad on the severely oxidized '70 Karmann Ghia. It takes a whole lot of time and effort to work out the oxidation. And I know that I'll then need to step down to FineCut and SMR (at least) to buff out the tiny scratches that HeavyCut introduces.
I'm also trying to work on a bad repaint on the clearcoated 75 Jag; tried the FineCut, and it did not even touch the bodyshop-installed rotary swirls. On the other hand, I kept the speed fairly low (3 to 4) and didn't bear-down with lots of pressure.
Fisher, the basic principle we use is to start with the least aggressive product/pad combo, and work up til you get the results you want. I don't think any of us would start with FineCut, unless it's really obvious (based on experience) that the paint is in such rough shape that it's not a risk; but I, for one, wouldn't be terrified to work up to it gradually.