Tier2 extending with new job, overstaying

I had my job interview last Wednesday and got accepted for the job. I have been on Tier2 visa for one year and it expires on the 15th. Now I am trying to get a certificate of sponsorship before the 15th so I can apply for my new visa. My employer informed me that I was not meeting the labour market test requirements as the job was not advertised for 28 days. However, as of the 7th of October all doctors are in shortage in occupation list. Does this rule apply to me as I had my interview before this rule?
My other question is, if I get my CoS after the 15th, and overstay by 2-3 days, am I still entitled to cooling off period?
Thank you
(Note: I had a similar question on a different thread but cannot reach my account)

However, as of the 7th of October all doctors are in shortage in occupation list. Does this rule apply to me as I had my interview before this rule?

It is the 6th of October not the 7th and it applies to the date the COS is assigned not your interview date. What is the SOC??

My other question is, if I get my CoS after the 15th, and overstay by 2-3 days, am I still entitled to cooling off period?

You must apply online and submit before your visa expires as if you don't, you cannot work or anything as you will have no legal status to be in the UK. Only exceptional circumstances with substantial evidence of why you could not apply in time would be accepted, i.e. you were hospitalised, serious accident etc etc. Not sure what you mean by 'entitled to cooling off'?? It is not an 'entitlement' as such, it is an immigration rule.

(Note: I had a similar question on a different thread but cannot reach my account)

Useful to state your other username