Praise be to Allah.
If you become junub, you have to do ghusl so that you will be able to pray. It is not permissible for you to pray if you are junub, and wudoo’ cannot take the place of ghusl, but if you cannot find water or you find cold water, and you cannot find any means of heating it, and you fear that you may be harmed by using it, then you can do tayammum.
As for not doing ghusl because you felt shy and embarrassed, that is not permissible and that does not make it permissible for you to do tayammum.
Praying when junub is a grave wrong, and you have to repent from it, because the Prophet (peace and blessings of Allaah be upon him) said: “No prayer is accepted without purification, and no charity (is accepted) that comes from stolen wealth .” Narrated by Muslim (224).
And the Prophet (peace and blessings of Allaah be upon him) told us that prayer offered without purification is one of the things for which a person will be punished in the grave. This has been explained in the answer to question no. 65731.
Your fear of Allaah should be greater than your fear of people, as Allaah says (interpretation of the meaning):
“Therefore fear not men but fear Me”
“Do you fear them? Allaah has more right that you should fear Him if you are believers”
The shyness that keeps you from doing obligatory duties is a blameworthy kind of shyness, rather it is weakness and incapability.
If you had gone out to the mosque and done ghusl there, and prayed, you would have been spared any embarrassment.
And Allaah knows best.