I want to know if it would be possible for a surgeon to "shut off" a person's senses without causing paralysis of any kind. The person would have to maintain their ability to speak and move, but not be able to hear their own voice or sense the physical world around them at all.
Even if it were possible to selectively destroy sensory (afferent) nerves alone, the individual would not be able to walk/move normally immediately afterwards, because there would be no environmental stimuli with which to orient one's actions.
Two of the senses people often overlook are balance (or spatial orientation) and proprioception (where the body parts are with respect to each other and the physical world). (Some people lump the two; it's easier to think of them independently, because they really do quite different things.)
Proprioception (and closely associated kinesthetics) is possible because of nerve signals from skin, tendons, muscles, and joints. This gives us the senses of limb position and movement, the sense of tension or force, the sense of effort, the sense of gravity.
Patients (lacking proprioceptive feedback) made large directional and amplitude errors when performing a reaching task without visual feedback....The errors were associated with differences in velocity and acceleration for movements in different directions.
Basically, without proprioception, you could not touch your little finger to your thumb without visual input, let alone actually perform more complex voluntary movements successfully.
The vestibular system exists through input from three semicircular canals located in the inner ear on each side. The canals are oriented at (basically) 90° angles to each other so that we can maintain our balance no matter which position our head is in. These are also connected to one's ocular reflexes. Together they're called the vestibolo-occular system, which functions as a reflex (VOR).
Three sensory systems, vestibular, visual, and somatosensory/proprioceptive, serve our sense of balance. Each of these systems provides unique sensory information to the brain where it is integrated (often in complex ways) to produce a final motor output to muscle groups for maintaining gaze, head and body posture, and controlling static and dynamic balance.
Without these systems, we would be completely unable to get or stay in a standing position. We certainly could not walk.
So, long story short, the answer is no. Not only is it surgically impossible, but if it were, you would not have a coordinated, working body left. Eventually even speech would be affected (as proprioceptors on the mouth are involved in speaking).
 Proprioception From a Spinocerebellar Perspective
 mpairments of reaching movements in patients without proprioception. I. Spatial errors
 Anatomical and Physiological Considerations in Vestibular Dysfunction and Compensation
The nerves for sensation do travel intimately close to motor nerves outsides the spinal cord. In these locations only disease processes can preferentially target a particular class of nerves. Diabetic neuropathy is a common condition causing this kind of peripheral nerve injury resulting in sensory loss. In the spinal cord, afferent (incoming) and efferent (outgoing) conduction pathways are segregated into bundles so a traumatic event can cause a selective deficit of sensation or movement of a region. Further up the chain (brain) entire functions are segregated so it is possible to create not only deficits of physical regions but of specific complex motor functions or sensory processes. Interestingly most of our senses are handled by cranial nerves that are not in the spinal cord. These nerves are generally purely sensory.