Your job offer letter is good for your application for permanent residence. That application however does not authorize you to stay and work in Canada - you do need a new work permit, and that is a completely separate application.
And yes, you will need a positive LMO to get a new work permit.
Note that to apply for PR as a skilled worker in the "arranged employment" category, you MUST remain in Canada on a valid work permit throughout the entire processing period, otherwise you will lose your eligibility.
So, two different applications / processes, but they are related. Note that with only 2 months left on your PGWP, you are running out of time very quickly. LMOs take time.
NO, it does not matter that your employment is in a different field than your degree.
The smart and well-educated can find all this information on the CIC website
