What is the ruling on someone who performed a second 'Umrah while in a state of ritual impurity (junub), out of shyness and embarrassment from his companions, and what is required of him now?

If you exited the first 'Umrah, the second 'Umrah requires a new state of ihram, and ihram is a pillar without which the 'Umrah is invalid. Entering the Sacred Precinct while in a state of major ritual impurity (junub) is an error, because a person in a state of junub is prohibited from entering theah mosque.

The ihram of a junub person is valid, but their tawaf (circumambulation) is not valid according to the majority of jurists, because ritual purity is a condition for the validity of tawaf. The Hanafis consider the tawaf of a junub person valid, along with the obligation of offering a sheep.

If you did not repeat the tawaf after purification, you remain in your state of ihram and must return to Makkah to repeat the tawaf and the sa'i (ritual brisk walking) after it, then exit the state of ihram. If you are unable to return, then you are considered muhsar (prevented from completing the pilgrimage) and must offer a sacrificial animal (hady).

If you entered the state of ihram for the second 'Umrah from inside the Sacred Precinct and did not exit to the hill (area outside the Sacred Precinct), you have omitted an obligation and must offer a compensatory sacrifice (dam), even though your ihram is valid.

Your prayer while in a state of major ritual impurity is a grave error, and you must make up all the prayers you performed in this state, and repent to Allah.