All perfect praise be to Allah, The Lord of the Worlds. I testify that there is none worthy of worship except Allah, and that Muhammad is His slave and Messenger.
Whoever has a wet dream must perform Ghusl (ritual bath) and it is not permissible for him to perform Tayammum (dry ablution) unless there is no water or that he fears being harmed by the use of water. The prayer is not valid except with this Ghusl.
Therefore, being shy should not prevent you from doing what Allah ordained you to do. Nonetheless, you may find another way that may help you not to be embarrassed, such as informing them of your situation because there is no shyness in religious matters and wet dreams are natural for human beings. You may also perform Ghusl in the amenities of the mosques that are close to you or you may look for another place to live.
Allah knows best.