The words used in the Psalm for “judge” is אֱלֹהִים “elohim” the word is often used for God but is not a specific title God. That is the word means, “ruler or judge” and God is certainly those things and the Torah attributes this title to God, but this attribution does not preclude the use of the word elohim in other contexts. For example Exodus 20 (the Decalogue) calls the pagan gods elohim, but never calls them YHWH. So even those things the Bible says specifically are not God are elohim in the Old Testament.
I think what Jesus was doing was playing on this word coincidence to prod His adversaries who were angry with Him because they knew full well that He was calling Himself God. However Jesus, knowing that they held the seat of Moses, were elohim and so could be called gods and so He was twisting the knife a little. Especially if we consider the character of Jesus versus that of the Pharisees and Judges (elohim) who devoured widows houses for gain. Truly One had earned to be called elohim as YHWH was called elohim and the others as the false idols who led men into Hell. Which is of course exactly what Jesus said they did, “You scour sea and earth to make one disciple and then make him twice as fit for Hell as you are yourselves.” Remember that?
However Psalm 82 does not call men יְהֹוָה YHWH or Jehovah that is exclusively a title appropriate to God Himself, and this is the title Jesus takes upon Himself. The LXX (the Greek translation of the OT) translates YHWH ἐγώ εἰμί “ego eimi” or “I Am.” This is the Name God gave to Moses in the wilderness the YHWH name and you will notice that when Jesus uses this appropriation for Himself the Jews gather stones to kill him because they understood full and well what He had just said.
So I think we can understand what Jesus meant by saying “Your Law says ‘you are gods’” and still not fall into the heresy of deification as the mormons did. Because of all the promises Jesus makes to us (we will be with Him, we will be as He is, we will live in Him forever, we will drink the water without price, we will eat of the tree in Paradise) He never says that we will become I Am because only God is.
If you have confusion about the distinction between is and becoming then I highly suggest you read Peter Kreeft in the Summa of the Summa his explanation of Thomas’ understanding on the being of God is extremely helpful if a little difficult. Perhaps someone else could recommend an easier text but for such a hard question maybe a little work is in order.