Impaired verbal communication, but normal reading and writing skills? Is there any cognitive disability or disfunction that impairs both speech production and listening comprehension, while reading and writing skills are at a normal level or better?
Just a reminder that Stack Exchange is not an appropriate place to get a diagnosis; a vague description of symptoms may indicate a variety of possible outcomes.  See a doctor instead.  That said:

One general term for disorders that affect some language modalities and not others, when the cause is psychological rather than physical (ie, not deafness, muscle weakness, paralysis, etc), is aphasia:


  To be diagnosed with aphasia, a person's speech or language must be
  significantly impaired in one (or several) of the four communication
  modalities following acquired brain injury or have significant decline
  over a short time period (progressive aphasia). The four communication
  modalities are auditory comprehension, verbal expression, reading and
  writing, and functional communication.


Aphasia is most commonly associated with stroke or dementia, and several sub-types involve both speech production and verbal comprehension difficulties, while not necessarily affecting reading or writing abilities.

Other diagnoses are also possible, as language disorders may consist of a combination of a variety of possible aspects of language ability, while not affecting others.  For example, children with both receptive (comprehension) and expressive (speech) delays may be diagnosed with specific language impairment (SLI):


  ... diagnosed when a child's language does not develop normally and
  the difficulties cannot be accounted for by generally slow
  development, physical abnormality of the speech apparatus, autism
  spectrum disorder, apraxia, acquired brain damage or hearing loss.


Though relatively rarely, it is possible for literacy to be largely unaffected by SLI.

Additionally, comorbidity is also possible, such as a combination of speech agnosia and apraxia, and if other symptoms are present, then yet other diagnoses would apply, such as autism, where again, it is possible for some language skills (such as verbal communication) to be impaired while others (such as literacy) are largely unaffected.

For a general overview of the field, see speech and language impairment and communication disorder.  A summary of technical terminology used to classify symptoms can be found here.