It is obligatory to do ghusl with water to cleanse oneself of janaabah, because Allaah says (interpretation of the meaning):
“O you who believe! Approach not As-Salaah (the prayer) when you are in a drunken state until you know (the meaning) of what you utter, nor when you are in a state of Janaaba (i.e. in a state of sexual impurity and have not yet taken a bath), except when travelling on the road (without enough water, or just passing through a mosque), till you wash your whole body. And if you are ill, or on a journey, or one of you comes after answering the call of nature, or you have been in contact with women (by sexual relations) and you find no water, perform Tayammum with clean earth and rub therewith your faces and hands (Tayammum). Truly, Allaah is Ever Oft-Pardoning, Oft-Forgiving”
“O you who believe! When you intend to offer As-Salaah (the prayer), wash your faces and your hands (forearms) up to the elbows, rub (by passing wet hands over) your heads, and (wash) your feet up to the ankles. If you are in a state of Janaaba (i.e. after a sexual discharge), purify yourselves (bathe your whole body). But if you are ill or on a journey, or any of you comes after answering the call of nature, or you have been in contact with women (i.e. sexual intercourse), and you find no water, then perform Tayammum with clean earth and rub therewith your faces and hands. Allaah does not want to place you in difficulty, but He wants to purify you, and to complete His Favour to you that you may be thankful”
And the Prophet (peace and blessings of Allaah be upon him) said: “ ‘Clean earth’ is the wudoo’ of the Muslim even if he cannot find water for ten years, but if he finds water then let him fear Allaah and let it touch his skin for that is good for him.” Narrated by al-Bazaar; classed as saheeh by al-Albaani in Saheeh al-Jaami’, no. 3861.
It is not permissible to do tayammum unless one has an excuse for not using water, such as none being available, or if using it will cause harm, because of the verse quoted above.
With regard to not doing ghusl because you feel shy, this is not permissible and tayammum is not sufficient in this case, and prayer is not valid if you do tayammum. You should put your fear of Allaah above 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 kind of shyness that prevents you doing obligatory actions is a blameworthy kind of shyness, indeed it is a kind of weakness and incapability. It has nothing to do with the praiseworthy kind of shyness.
In order to make you feel less embarrassed, you can do ghusl at times when they will not notice it, such as before dawn, or you can use the water in such a way that they will not hear it and no one will know that you are doing ghusl, and so on, which will help to conceal the things that may be causing you to feel shy.
And Allaah knows best.