The words you came across in the poem refer to Middle English, if i'm not mistaken. William Shakespear's sonnents are abundant in suchlike. "Thee" stands for "you", "thy"-"your", "cometh"-comes, "doth"-does, "hath"-has etc.
So, if translated into Modern English you get the following:
I said: "I am grieving (am sad) because of you"
She said: "Your grief is coming to an end"
I say at once: "Be my moon" (meaning be my only one as nothing can compare with the beauty of the moon at night)
Not sure about the translation of the last line
Why do you think the man has a grief for her? Because he is head over heels in love with her. And she is either refusing to become his betrothed, or has no feelings for him at all.