You are correct for Jesus did, by his sacrifice, make it possible for us to have everlasting life. Even those now dead will soon arise as Acts 24:15 says "There is going to be a resurrection of the just and the unjust." So what the Bible reveals in the Old Testament regarding what can happen to the soul is correct. Otherwise there would be no need for a resurrection, a bringing back to life from the dead, the opposite of life.
Note how our master Jesus cleared things up regarding what happens at death. He does this at John 11:11-14 where it is recorded:
(John 11:11-14) "After he said these things, he added: “Lazʹa·rus our friend has fallen asleep, but I am traveling there to awaken him.” 12 The disciples then said to him: “Lord, if he is sleeping, he will get well.” 13 Jesus, however, had spoken about his death. But they imagined he was speaking about taking rest in sleep. 14 Then Jesus said to them plainly: “Lazʹa·rus has died"
Being a master teacher, Jesus knew that in verse 11 when he said that Lazarus has fallen asleep, his disciples would logically think he was referring to normal sleep as verse 12 shows. But then Jesus opens their eyes to what death is in verse 14. Teaching in this way helped the disciples to see that death is likened to sleep, a sleep that only Jesus can awaken us from. This helped his disciples make the connection between what had just taught and what is stated in the OT at Ecclesiastes 9:5, 10 and Psalm 146:3,4. The dead truly are conscious of nothing at all but sleeping in death. See how the OT and the NT agree helps understand what the Bible states at 2 Timothy 3:16, 17: "16 All Scripture is inspired of God and beneficial for teaching, for reproving, for setting things straight, for disciplining in righteousness, 17 so that the man of God may be fully competent, completely equipped for every good work."
Yes, both the OT and the NT agree regarding what happens to a person (soul) at death. Yes, everlasting life is promised and will be realized once Jesus starts to fulfill Acts 24:15 "resurrecting the just and the unjust". But for now those now dead are sleeping as did Lazarus was before his resurrection.