i dont know about the new rule or changes is the current rules, but.....
its not necessary to cancel the visa of your previous country of residence to get the new visa of a different G.C.C country, but.......
if you plan to go back to the same country(maybe after couple of years) or try to get some police clearence certificate( maybe for cannadian, australian immigration) they will not issues the same as the visa was not cancelled properly.
what i suggest is,ask the guys to get the visa of the new planned country and then resign by giving one month notice and follow the procedure for cancelling the visa, then proceed directly to the new country.
hope this will help