> rotate through all available Azure IPs, ... They should come from the same IP address.

I would guess that this is intentional, intended to prevent IP level blocks from being effective. That way blocking them means blocking all of Azure. Too much collateral damage to be worth it.

It is. There are scraping third party services you can pay for that will do all of this for you, and getting blocked by IP. You then make your request to the third-party scraper, receive the contents, and do with them whatever you need to do.