Yes, as I said, once you straightway add your ESXi host to new vCenter, it will give you one warning as shown in below snapshot:
If you continue above UI workflow, ESXi host will be disconnected from older vCenter automatically.
Other way of doing the same is as follows:This is recommended way:
1. First set your DRS in manual mode, disable HA on older cluster.(old vCenter): But if same cluster has other ESXi host those you are not moving /owning, then this step need to be skipped as it will have impact on VMs on other hostss (due to HA disablement)
2. Disconnect ESXi from older vCenter.
3. Add the same ESXi host on new vcenter cluster.
But in your case, as you do not have control on older vcenter, first way is the only way you will have.
On EVC, as I already specified, please study well how EVC works & go ahead with the same.