To intentionally omit a serious sin during the sacrament of Confession is a sacrilege, so it invalidates the confession. Those sins must be confessed again, as well as the sin of purposeful omission of a mortal sin.
To remedy and be forgiven, the person should go to a priest asap. He has a good conscience to be concerned - that's God's grace.
Regarding embarrassment in the confessional:
...embarrassment is a terrible feeling. Rather than being afraid of it, embrace it. The embarrassment is a good thing. It is guilt, and no matter how much the culture tries to get rid of it, guilt will never go away until you confess your sins. Your guilt is a gift from God; it is His grace working in you. It encourages you to change your life and come back to Him. Once you have confessed your sins, you will feel like a truckload of bricks has been lifted off of your back. It is liberating, refreshing, and you will probably walk out of the confessional with a smile on your face just from the feeling of relief.