I have found with my Murkur 38c that like you my skin dries very quickly so I try make sure I really soak with hot water the area I'm going to shave well and apply Shaving gel or whatever you use nice and thinly. If you have long stubble a fresh blade is ideal otherwise a blade normally last me around 3 shaves. But I use my Murkur for shaving my head which is a larger area than ur face. I very rarely cut myself and its normally my fault when I do.
It's more than likely your soap/foam than the razor itself. This is coming from someone with dry skin naturally.