Sounds like they have a bit more wasta
than you, to say the least. At least they don't work for the Ministry of Labour. You have several choices then.
1. Go to the Ministry of Labour anyway and hope for fair treatment. Or at least try to phone their helpline and explain everything, then ask them what you should do. Given the wasta involved I can understand you might have legitimate concerns about this, but I don't know if there's any better options.
2. Find a lawyer in Dubai
to help with your case, but that's probably going to be expensive. Sometimes legal firms will at least give you an initial consultation cheap or free, and suggest some options.
3. Try filing a court case, but unless you contact the MOL first, and/or hire a lawyer, that's probably going to be a very difficult process, and risky given the wasta situation.
4. Try and figure out what your sponsor/employer wants you to do to be released from their service and play it their way (but I expect that's not going to be anything helpful for you, especially if they simply don't want to cancel your visa).