I think you have a right to be frustrated because your plans changed, but not to the extent you should be angry or upset with your mother. She has a right to come and go as she pleases. They do you a favor by giving you the car when they are not home. You have options. You can rent a car if you need to, or borrow from someone else if you must. Plus, your mom is asking if you need the car. Maybe she is willing to let you continue borrowing it. If you don't want her to know your plans, fine. Just tell her no worries and get the car back to her. Or, be direct and ask her if it's okay if you keep the car but don't give her your whole itinerary. Just say you need it for a trip. Also, you say you keep 'their cars while they are away" so do they have more than one? Maybe your mom can use the other car while you keep this one.
I'm sorry but I don't see this as a situation that requires any emotion beyond some frustration or disappointment, so perhaps your depression is clouding your judgement. You don't say you will lose money if the trip is cancelled or anything like that, you say you just don't want to tell your mom because she'll "involve" herself. I'm not sure what that means but you should be able to tell her you need the car and see if something can be worked out. Otherwise, reschedule or find a rental car.