You can get a shavette cheaply enough, which is the same as a cut-throat but uses half a safety razor blade. I had the same problem as you describe and switching to a shavette solved it. I do nick myself occasionally, but then I used to get cuts off modern safety razors too - usually when they intersected with ingrown hair bumps. I think I'm simply too hurried in the mornings.
You can get a shavette cheaply enough, which is the same as a cut-throat but uses half a safety razor blade. I had the same problem as you describe and switching to a shavette solved it. I do nick myself occasionally, but then I used to get cuts off modern safety razors too - usually when they intersected with ingrown hair bumps. I think I'm simply too hurried in the mornings.