This Psalm repeatedly references David's sickness- 38:3, 5, 6, 7, 8, 10, 17. He is suffering intensely.
He is suffering because of his sin- 38:3, 4, 5, 18.
This is not to say all suffering is directly connected to our personal sin. The book of Job shows that is not true. However, we cannot say that there is no connection between personal sin and suffering either. David experiences it right here.
His enemies add to his pain- 38:12, 16, 19, 20- while his friends stand far off- 38:11.
Jesus experienced the suffering the author of Psalm 38 experienced:
His friends forsook Him- Matt. 26:55-56, 58; Luke 23:49
He was beaten so badly there was no healthy part of His flesh- Isaiah 52:14; 53:5; Matt. 27:26-31.
He enemies laid traps for Him- Mark 3:6; Luke 11:54; 20:20
Jesus was like a deaf man- Matt. 27:13 and a mute man- Matt. 27:13-14
Jesus was not like the Psalmist in the sense that He never sinned- Hebrews 4:14-16; I Peter 2:22.
Jesus is able to carry away the burden of our sins described in Psalm 38:4- Matt. 11:28-30. Jesus can save and forgive.