I changed icloud password now cannot access mac

dues to email address being hacked, i decided to take action, what a mistake.

i changed my email address for apple, and then on mac was not letting my into the icloud, so had to change the password, so both ID and password now changed. however, now imac will not let me in as with the little shake suggesting i am putting in wrong password, i am not

there is no way to get the serial number, so am finding it tough to get support, and i seem to be going round in circles not getting any where

this is a new ma, so new, was very excited, now just very upset that i can't even log because of these links

help? thanks, Dax