lay my case ... fill my mouth

These two lines mean the same thing and emphasize Job's desire to explain his situation to God. (See: rc://en/ta/man/jit/figs-parallelism)

fill my mouth with arguments

Here "fill my mouth" refers to speaking. Alternate translation: "I would speak all of my arguments" (See: rc://en/ta/man/jit/figs-metonymy)