Change of job at the time of existing work contract is expiring

Hi,

I am on a limited period ICT work permit which is expiring by mid Aug 2021. I've already received letter from Ausländerbehörde to apply for extension. My current employer is yet to provide the necessary documents such as new extended contract, Arbeitsgeberbescheinigung, etc. This might take as late as first week of July 2021.

Meanwhile, I have got another job offer from a local employer. I am yet to get the contract. Also they are insisting me to join from Sep 2021.

Now my questions are as follows.
1. Can I change job while my existing contract is expiring and so I need to apply for extension?
2. Current employer will insist on serving 3 month notice period before relieving me. If I have to join by Sep 2021, I have to resign already by June 2021. If I do that, current employer will not provide document for work permit extension and so I will be forced to leave Germany by mid Aug 2021.
So, is it possible that I get contract from new employer effective from 1st Aug 2021 and apply for new Ausweis?
Note: If this is possible, I can get temporary visa and continue to stay in the country or re-enter in case my current employer asks me to leave immediately.
3. To be on safe side, I am thinking of applying for extension of my work permit under my current employer and while it is being processed, I can approach Ausländerbehörde with another application for change of job. I can negotiate joining date with new employer as Oct 2021 instead of Sep 2021. Is this even possible and accepted by Ausländerbehörde?
4. Can I discuss my above situation with Ausländerbehörde and take their advice? Will they understand and help?

Kindly suggest.