You're right, of course, that the Hebrew Bible makes a distinction here. The whole idea of the Messiah is that he is "the anointed of the LORD"--i.e., "the anointed of YHWH."
Recognizing that "the LORD" in the KJV means YHWH, we can substitute YHWH in all passages that use "LORD."
One such passage is Isaiah 61:1:
"The Spirit of the Lord GOD is upon me; because YHWH hath anointed me to preach good tidings unto the meek; he hath sent me to bind up the brokenhearted, to proclaim liberty to the captives, and the opening of the prison to them that are bound"
Jesus applied this passage to himself at the beginning of his ministry in Luke 4:
16 And he came to Nazareth, where he had been brought up: and, as his custom was, he went into the synagogue on the sabbath day, and stood up for to read. 17 And there was delivered unto him the book of the prophet Esaias. And when he had opened the book, he found the place where it was written, 18 The Spirit of YHWH is upon me, because he hath anointed me to preach the gospel to the poor; he hath sent me to heal the brokenhearted, to preach deliverance to the captives, and recovering of sight to the blind, to set at liberty them that are bruised, 19 To preach the acceptable year of YHWH. 20 And he closed the book, and he gave it again to the minister, and sat down. And the eyes of all them that were in the synagogue were fastened on him.
Jesus thus declares himself to have been anointed by YHWH.
Another such passage is Psalm 110:1:
"YHWH said unto my Lord, Sit thou at my right hand, until I make thine enemies thy footstool."
Jesus also used this passage in relation to himself:
42 Saying, What think ye of Christ? whose son is he? They say unto him, The son of David.
43 He saith unto them, How then doth David in spirit call him Lord, saying,
44 YHWH said unto my Lord, Sit thou on my right hand, till I make thine enemies thy footstool?
45 If David then call him Lord, how is he his son?
Jesus thus presents David's reference to "my Lord" as referring to the Messiah-- to whom YHWH said, "Sit thou on my right hand...."
To argue that Jesus is sitting at his own right hand would seem every bit as silly as the "Trinitarian" [sic - modalist] views Latter-day Saints often refute, which would have Jesus sitting at his own right hand when Stephen sees the Son seated beside the Father. YHWH--as referred to here--is not the Son/Messiah but the Father.
However, Jesus also identifies himself with YHWH, as in John 8:
57 Then said the Jews unto him, Thou art not yet fifty years old, and hast thou seen Abraham?
58 Jesus said unto them, Verily, verily, I say unto you, Before Abraham was, I AM [YHWH].
59 Then took they up stones to cast at him: but Jesus hid himself, and went out of the temple, going through the midst of them, and so passed by.
So, rather than the simple equation of Elohim with the Father and YHWH with the Son, it would appear that both the Father and the Son are identified in scripture with YHWH.