Emotion

อารมณ์

• The word cry has special usage here. Cry is both (transitive & intransitive) verbs and (countable & uncountable) nouns. Cry can be followed by a noun or noun phrase. For example:
     • Cry crocodile tears: to hypocritically show sorrow; insincere tears
     • Cry one's eyes or heart out: to cry excessively or inconsolably
  In this song's context, I'll cry me a river, cry is followed by a direct object me then an indirect object a river meaning the persona will cry so much that her tears can be as abundant as a river. The relative pronoun clause that follows--that leads to your ocean--wonderfully and poetically corresponds to the preceding one, creating a unified, beautiful figure of speech called hyperbole.
• The (informal) phrasal verb fall apart (as in You never see me fall apart): If you say that someone is falling apart, you mean that they are becoming emotionally disturbed and are unable to think calmly or to deal with the difficult or unpleasant situation that they are in (synonym = crack up). So the Thai word choice should be as emotionally devastating as the English original. It doesn't mean just เจ็บปวด but more than that. Hence, รานร้าวพังภินท์ is the best word choice I could find.
• The phrasal verb have got to: In the first one--But you've got a part of someone else--have got means 'to have,' and in You've got to find your shining star, have got to means something is necessary or must happen in the way stated (synonym = must). So the Thai translation should be จะต้อง.