Will any cheap ozone generator do the job?

Putting aside the obvious product recommendation part of the question here, an ozone generator will get rid of the the moldy smells you are encountering. Ozone destroys bacteria and living stuff, so will destroy the microbes which are causing the smell in the first place.

There are decent ozone generators on the market which do a great job at a reasonable price (~$60 here in the States). These are powered by household current and not your car battery. I'd have no idea how well the 12v versions would work, but would suggest they are not going to work as well as the household powered ones, won't have the capacity of them, and run the risk of draining your car's battery.

To use one correctly, you should ultimately follow the directions. With that said, basic instructions are to leave the car sealed up with the ozone generator running. Run it for 30-60 minutes, then allow the ozone to sit in the vehicle while still being sealed up for another 2-3 hours. You can find versions with a timer on them which will automatically shut off after a period of time so you don't have to worry about it. Once this is done, open all the windows and allow to air out for a 1/2 hour or so. This will completely clear the car of any ozone, so you shouldn't have a fear of it causing you health issues. You do not want to breathe the ozone for any length of time because it will destroy lung tissue with extreme exposure. Yes, minimize your exposure, but minuscule amounts are not going to hurt you. Just open the doors to let the majority of it out, then open the windows and leave them open for a while (that way you don't drain your battery by having the dome lights on). Having limited contact with your skin won't cause you any issues. Remember, there's ozone all around us to differing degrees all the time.