Imran, I m from Kuwait, so I can only advise as per Kuwait Labor Laws, which may be similar to other gulf countries,
If you are on the employer's Commercial Visa then cancelling your Work permit (Izn Amal) also cancels your residence status in the country. if you have completed the contract period with the firm, you can request for release from your current employer so that you can join another firm.
As you mentioned that you have resigned from the job, served notice period also, however sponsor employer is harassing you by not giving you a release, then you can approach relevant ministry of labor in the country, I dont know what is the exact name of the ministry, but you can research and find out. Also make sure you read the contract carefully, because usually in work contracts, not to work with company's clients or compettitor clause is there. however you can double check. if there is no such clause mentioned, then you can appraoch ministry.
However make sure, you have completed the contract period with the firm, served resignation and notice period, check any papers you have signed which mentions not to join compettitor clause, also check if you have signed any training contract which has lock in period.
If your work permit is cancelled, you may be on a temproray residence visa, if your new employer really wants you can discuss the issue with new employer and seek a new VISA and re-enter Dubai.
Lets us know, what was the outcome.
26th July 2011 From Kuwait, Salmiya
14th June 2015 From India, Mumbai