Can childhood propioceptive disorder behaviors continue into adulthood?

Yes: Much as with attention-deficit, which was thought to not happen in adults, there is neither a biological reason, nor any evidence that sensory integration / sensory processing disorders like disorders of proprioception should vanish when one becomes an adult. Thus these disorders can definitely persist into adulthood.

Yes: Yes, but most seem to stop in late childhood in my experience.

If difficulty with: proprioceptive/kinesthetic sense still impacts handwriting & other motor functions at age 12-13, it is likely to be from changes in the parts of the brain that regulate this, even if no damage is seen on an MRI. The American Academy of Pediatrics does not recognize Sensory Processing Disorder as a separate diagnosis, as it is found in a variety of neurodevelopmental disorders that are life-long.