It seems like the higher end razors like the Mach 3 do give a closer shave, but I don't understand the point of the pivoting head they all seem to have. On more than one occasion it's given me a pretty deep gash with 3 stripes of blood going around corners. What am I doing wrong? Are there any good razors that don't have a pivoting head?

You don't use much pressure and you never ever let it slip sideways.

And they pivot so they can keep in better contact with your skin without you having to be precise in the angle of the handle.