Tier 2 General - Job Switch

Hi,
I have a Tier 2 General visa valid from May 2016 till May 2021. I have a wife and 2 kids who are on my dependent visa. I have a job offer and the new employer is willing to sponsor my visa. What's the process to switch the sponsorship?
I believe that the new employer will require a new certificate of sponsorship and once the employer gets the new certificate of sponsorship, I will have to apply for a new entry visa and then get a new BRP. Is it correct?

Should I resign from my current job and start the notice period (which is 8 weeks) now or wait for the new employer to get the new certificate of sponsorship? Once the new employer has the certificate of sponsorship, how long do I/employer have to apply for the new entry visa?

Will I have to pay the health surcharge again or the one I paid for the previous visa would work?

Also, I will have to change the visas of my dependents by paying visa fees, health surcharge as well or is it just me who needs to change the certificate of sponsorship and the visa?