Random Orbital will NOT harm paint! High speed buffers like the Milwaukee or Makita (POS) will burn paint.
I have seen damage done with a RO. Older paint and problems with oxidation make it more likely.
To the OP, always move front to back, never circular motion. Set it on 5-6 for speed. I find using a spray bottle with some distilled watter in it can help. You need the polish to stay moist for a minute or so to let it cut the paint correctly.
I would also recommend doing a test area to see just how aggressive you need to go. I would NEVER use Scratch-X on a RO or on the whole car for that matter. It is for spot work, by hand. In general with whatever system of polish you are using you need to work from the finest polish/lightest cut to more aggresive, only going as far as you need to remove the damage to the paint. This is why I recommend a test patch. You can then know just how far to go without polishing the whole car, only to learn you need to go heavier cut to get your swirls/scratches out. Scratch-X is just a step below the professional 3M cutting compounds as far as degree of cut so you are skiping over all true polishes to use this. You will then need to "back out" to the lightest polish from there.
Here is a cut section out of my long detail word doc for you about scratch removal:
Process of Scratch Removal (shallow – not through the clear or color coat)
The key to safe scratch removal is working with the least abrasive compound that is necessary to solve the problem. Whatever product you use, you want to work it on the surface for a brief bit, buff off the product, examine and repeat as necessary. If 3 or 4 rounds of this doesn’t solve the problem, it is time to move to a more aggressive product. Once you get the problem solved (or minimized to your satisfaction), you move backwards through the progression of products you just used… using less and less abrasive products to buff out the scratches caused by the last one.
Again, a full write-up of scratch removal could very easily morph into a huge document. Do your research and work slowly and carefully.
Here are my products (in Abrasive order):
Least Abrasive
• Paintwork Cleanser (Sonus makes a good one)
• 3M Swirl mark Remover/Sonus SFX-3
• 3M Finesse It II/Sonus SFX-2
Sonus SFX-1
• Scratch-X (Meguiars)
• Fine-Cut Rubbing Compound
• Medium Rubbing Compound
• Rubbing Compound (I leave this one to the professionals and won’t let myself buy any because I’d probably do some damage)
Most Abrasive
Hope that helps.