Verb-고 싶다’s negation should be either Verb-고 싶지 않다 or Verb’s Negation + 고 싶다. So 말하고 안 싶다 is incorrect.
말하다 is a verb that has “Noun + 하다” form. So following the corresponding rule for such verbs, so you can negate it either 말하지 않다 or 말 안 하다. In addition, 말하다 is also an exception verb where putting 안 right before the full verb is allowed. So 안 말하다 is also a correct negation.