I spent a lot of time looking at the various budget cordless dual action polisher. I found that most of the 12 volt ones are more for applying wax and maybe some light polishing. The exception to this would be the Batoca S2 dual action polisher which comes with two 12 volt 5 amp batteries. I've been using this polisher for a few weeks, and it does a good job. The pad doesn't tend to stall out much at all which is really important to get any kind of correction. It's suitable for light paint correction on speed 5 or higher. Another one to look at is POPULO brushless buffer on Amazon which has more power but only comes with one battery. I looked at the Kobalt polisher but about half the reviews said that the pad stalls out with just barely any pressure.
A good budget combo would be the Batoca S2 polisher and the Turtle Wax Ultimate Paint Correction kit which comes with the Pro 1 & Done compound, 3 Lake Country CCS foam pads and one Buff and Shine 50/50 Uro-fiber pad. It's being sold on the official Turtle Wax site for $42 which is a great deal. Basically, you would want to start off with the least aggressive pad on a small test section. Keep testing until you get the correction that you're looking for. One thing that isn't always mentioned on most of the paint correction info that can be found online is how hard the clear coat is on the car that your working on. If it's soft, you can get a lot more correction on a less aggressive pad. If it's a really hard clear coat, then you might need to use a microfiber pad.