Note: I am not an expert, everything below could be wrong.
“But does the sense of touch saturate?”
Yes
“And if so, how?”
What you are calling “saturation” is known as neural adaptation or sensory adaptation.
The activity of senses is broken in to two distinct categories, “sensation”(sometimes called “reception”) and “perception.”
Sensation occurs when an external input causes a change in the electrical potential of the membranes of cells which are specialized receptors. This change in potential sends an electrical signal to the specific areas of the central nervous system (for touch, this area is the somatosensory cortex) which processes the signal and creates a perception of that external input. The neurons in the CNS have action potentials (known as spikes) which encode specific information in the neuron and send that information to other neurons by varying the frequency and pattern of these spikes. When a neuron is activated by sensation the relevant neurons become excited and have a high frequency of spikes, but if the input causing this excitation continues the frequency of spikes begins to drop. It is currently a matter of investigation as to why this happens, AFAIK, and the partial explanations available are a bit beyond my ken.
As the frequency of spikes drops the perception of the sensation which is activated goes away.
“And does this explain why we wear clothes?”
No, I shouldn’t think so.