If by safety razor you mean DE razor, then yes. Cartridge razors encourage you to press down for a close shave, which increases skin irritation. A good DE razor with a sharp blade, it's easier to shave with a light touch. The tradeoff is you need much better shaving technique. Compared to a DE, it's pretty much impossible to cut yourself with a cartridge razor.
The biggest factor with skin irritation is lubrication though. If you switch to a high-quality shaving soap (not foam/gel), apply it with a good shaving brush and make sure you shave after a hot shower, you will probably see a big improvement even using a cartridge razor.
Using a good non-alcohol based aftershave will also help with your ingrowns. I'm prone to them as well and I find the only way to really avoid them altogether is to exfoliate and moisturise regularly, annoying as that is.
Cheers, I'll invest in one soon. I think part of it is also that my skin isn't used to wet shaving any more. Even with a proper technique I'll probably have a few issues for the first few months.
Have been using a good water-based, non-greasy moisturiser for over a year now. Works wonders. All that sickly smelling stuff they sell at the supermarkets should be banned...it makes the irritation worse from my experience.
One thing I've done in the last few months is apply a hot face washer for around 2 minutes before a shave, then a cold one after for the same period. Wish I knew that trick earlier, has saved me a lot of grief.