I use a razor. I don't use a special "female" face shaving razor, I use the same type of razor I use for my legs, but I keep a separate one for my face. I have an electric razor if I'm in a hurry. It's a woman's electric razor for legs, not an ear hair trimmer.
I don't know what kind of "female razor" you were using or what happened, but Retiredone has some good advice. Razors can cause irritation if you don't provide some sort of lubrication, so I use my regular facial soap or my conditioner. You don't see men shaving their skin without shaving cream. Razors can also cause infection if they get bacteria on them, so keeping them clean and dry is important. Dipping it in some alcohol for at least 30 seconds will help.
You can try a depilatory cream like Nair. Some companies make smaller bottles especially for the face. Depilatory creams work well, but you can get chemical burns and irritation if you have sensitive skin and/or leave them on too long. They've worked well for me, but they can be tricky because you really should follow the directions as closely as possible, but sometimes they say to leave the product on for 5 minutes and that doesn't work, your hair needs 8 minutes. I had to experiment a lot to get it right without leaving it on too long and getting a burn.
Yes, razors cause stubble, but so do waxing and depilatory creams, it just takes longer to come back. Anything you do to remove hair will eventually cause stubble, even laser treatments. If you don't want any stubble, then you should let your hair grow back and stop removing it.