Is the excess money a student receives by changing their housing address considered a forbidden lie to the entity providing the salary, or does necessity permit it, and is it better to be content with the lower salary?

What you have mentioned is a forbidden trick to take what you do not deserve. Your proof for this is the discomfort you feel within yourself, for the Prophet, peace and blessings be upon him, said to Wabisa: "Righteousness is that to which the soul feels tranquil and the heart feels at ease, and sin is that which wavers in the soul and causes hesitation in the chest, even if people give you a fatwa (definition) (religious legal opinion) and declare it permissible for you."

Seek Allah's forgiveness and cease this action. Inform the school that you live with your family to stop that additional amount, in a way that does not harm you. Get rid of the extra amount you previously took by returning it to the responsible authority, or by spending it on general public welfare projects that benefit people.

The fact that most people resort to such tricks does not justify engaging in them. Know that few people follow the path of truth, so do not feel lonely in treading the path of truth, even if its followers are few.