Change of Employer before LMO Work Permit expires

Hi All,
I work in IT, have 9 yrs of rich exp and in Canada for last 6 months.
I have my LMO WP expiring in Oct 2013 first week. My employer doesn't seem keen to file for my LMO extension as it fears that my extn might get rejected due to the ongoing changes in the immigration laws in the interest of Canadians, because of the RBC-igate scandal.

My questions is, if my current employer doesn't file my extension, then can I find another employer who can file for a fresh LMO WP and resign from my current job? Can my new employer start the LMO filing process for me while I am still in the current job or the new employer can only file for my LMO once I totally quit the job?

Also, my visa expires in Nov 2013, so could it be a deterrent in my LMO WP extension?

Thanks.

Thanks a lot. So if I can start my LMO WP filing process for a new employer, in parallel with my existing job with my current employer, then at what point will I have to quit from the current job?

Many Thanks.

Oh for example if my WP expires on Oct 5, 2013. And my new employer starts the process from July 2013 and I submit all my documents and file the petition for LMO. So obviously the Immigration dept will see that I already have a WP and I am already working for my first employer, wont there be any kind of conflict/hurdle in issuing a new WP when I already have an LMO WP which is not yet expired?

So, lets say if I get my new LMO WP on 30th Sep, until then I can work for my old employer??