The anxiety you feel is not something to fear or hide from. It is the honest cry of a heart that knows its need, and that is exactly where salvation begins. The good news is that what you are seeking is not far off or hard to obtain. Salvation is as near as your own mouth and your own heart. If you will confess with your mouth that Jesus Christ is Lord and believe in your heart that God has raised Him from the dead, you will be saved.
You spoke of needing to be saved, and you are right to see it as a need, not an achievement. This is the great mystery that so many stumble over: we do not work for this salvation; we receive it as a gift. Even the faith to believe is given to us by God, so that no one can boast. The cross of Jesus Christ is where the righteous demand of the law was fully met. The penalty that rightly belonged to the sinning soul fell upon Him. He became the captain of our salvation, made complete through His own suffering, so that He could bring many sons and daughters into glory. There is no other name given under heaven by which we must be saved. He is the chief cornerstone, and all who trust in Him are built into a dwelling place for God.
Do not let the simplicity of this cause you to hesitate. Now is the accepted time. Now is the day of salvation. The grace of God that brings salvation has appeared to all people, and that grace is tied inseparably to the forgiveness of your sins. To submit now to the Lordship of Jesus Christ means eternal life. Later, every knee will bow and every tongue will confess that He is Lord, but that will be a confession of judgment, not of salvation. Today, that confession brings life. Believe in your heart that God raised Him from the dead, and speak it out. You will be saved.
The struggle for assurance often comes when we look to our own performance rather than to the finished work of Christ. The true mystery is Christ in you, the hope of glory. You are not saved by a ritual, a tradition, or a careful observance of spiritual duties. If you were trusting in those things, Christ would profit you nothing. You are saved by faith in Jesus Christ, and that faith will realign your entire life. But the first order, the thing that brings peace, is to stop looking at your own ability to be righteous and instead receive the righteousness that is yours through faith. The righteousness of Christ is counted to you. You are clothed in Him.
So, in this very moment of anxiety, let your heart be comforted. You are not called to strive for a salvation that might be just out of reach. You are called to confess the Lord Jesus and rest in what He has already done. The work of salvation is His. Your part is to believe and to receive. The peace you long for is not found in perfect feelings, but in the Person who has promised that whoever comes to Him will never be cast out. Walk in that today.