Have you tried codeine?, it's an opiate so not NSAID and it's available in a variety of strengths so with a bit of trial and error on the dose you can work out your tolerance so that it doesn't make your drowsy so you're ok to drive. I have the same problem with work as I drive 60 miles each way, but codeine is ok if you get the dose right and will definitely help you more than plain paracetamol.
If you get it on prescription the dose is normally 60mg (2x tablets), I wouldn't suggest taking that and driving as it will most likely make you drowsy and a bit of a zombie (the 60mg tends to send me straight to sleep or very spaced out!) but the version you buy over the counter at the pharmacy is normally 16mg and mixed with paracetamol (sold as co-codamol) and this should be ok for you to drive.
I tend to start off by taking just 1 of the lower dose (so 8mg codeine) and take it with 1 paracetamol so still get the full dose of that and then if I start to struggle during the day I then go up to the 16mg codeine dose, and then up to taking one of the prescription strength (so 30mg codeine) if it's a bad day and I can still drive on that, but I preferably avoid the 30mg if it's within a hour or so of driving home - not worth the risk! I've come to learn that if I need any more pain relief than that then I probably should be at home instead!