If you use a hair dryer, mount it on the wall to leave your hands free. The bracket should swivel so the dryer can blow in any direction. Hair washing is easiest in the shower. However, if you bathe in a tub, a hand sprayer is useful for rinsing. Soap mitts are inexpensive and can be found at most stores where bath supplies are sold.
Electric razors are easier to use and safer than blades.
A bracket or clamp can be rigged to hold a razor firmly to a counter or wall while you move your face against it, or a Velcro strap holder can be used to secure the razor in your hand. A nailbrush can be mounted on the edge of the counter or sink with suction cups. Attach a nail file or emery board to a piece of wood or tape it to the countertop to stabilize it for one-handed use. You can attach a nail clipper to a piece of plywood to give you better control, or increase the leverage by lengthening the handle as shown in the illustration. Larger clippers for trimming toenails may be easier to use on your fingernails.