You said it is a medical problem, but you didn't say what is being done to help you with it. You and your mother should go talk to the doctor and see what can be done to help you with it and perhaps have him recommend someone to talk to that will help you not to feel so bad about yourself.
There are products on the market now that will keep you from soiling your sheets at night. They are kind of like the pull-ups that young children wear. Hiding your sheets in a box and not telling anyone isn't helping you learn to live with the problem.
A medical problem is hard to hide from people that you live with, and with understanding from them that it is something you are getting help with, there shouldn't be a problem with them calling you names or telling others.
Talk to your mother and the doctor about these pull-up type night pants to wear to keep your sheets dry and find out what the medical problem is and ask what can be done to help you with it.