Can I still change employer in the new permit policy in Shenzhen?

I have worked in Zhejiang province for the past 3 years and I am now looking forward to work in an IT company in Shenzhen. But the visa and permit rules are changing from November 1st. I would like to know if I can still change employer or transfer my visa to the new employer in Shenzhen.
Secondly, when I came to china in 2012, i was not told to bring any criminal records certificate and also wasn't told to authenticate my degree certificate then. But it seems the new policy they are demanding all these stuffs, Is possible to get the permits in Shenzhen without providing criminal records or authenticated degree certificate because I have been living in China since 2013.