Dear , Qatar rules are not much employee friendly. Since, you have got sponsorship, no matter it is from the company you work for or from some other person who has given them permission to hire people under his sponsorship. They know how to tilt laws in their favor. so, On sponsorship change front you cant do anything. Under present rules in labor law, NOC is very important before you join another company or change sponsorship. If you do, your present Kafil can sue you and get you out of Qatar in few hours and you can be fined as well.
it is better you talk to your kafeel in a nice way and see whether they agree to provide you NOC or not, if yes, you won if no, it is better to wait till you contract finishes. Visa can be renewed even pataka has 3 months grace period. So talk to kafeel.