I want to build a NEW vCenter server and consolidate 3 existing vCenters into this NEW vCenter. Can I just disconnect the hosts from their OLD vCenter and add it to the NEW vCenter? Can I leave the VMs running on those hosts while I do this?
You can leave the VMs running, but depending of your network configuration (if using vSS or vDS) you will need some additional steps, but the good news is that VMware have it documented for you, see:
Moving a managed ESXi/ESX host from one vCenter Server to another vCenter Server (1004775) | VMware ...
Moving an ESX/ESXi host with vDS from one vCenter Server to another (1029498) | VMware KB