Do I have to wait 60 days to transfer website from hosting service to another hosting?

"I believe the ""60-day hold"" rules you are thinking about refers to ICANN's policy on transferring domain names between registrars. Not to changing your DNS settings which would allow you to move your website to another host.

Policy on Transfer of Registrations between Registrars

So, unless you have some sort of minimum service-length contract with the web host, you should be able to move between them freely."