My VMware ESXi is licensed for one physical CPU. If I decide to upgrade my server with an additional CPU, do I need to buy a new license or I can upgrade the existing one by paying only the difference?

Another CPU means another license.

Can I use Vmware HA and DRS with the existing license?

vSphere Enterprise Plus supports HA and DRS, but you will need all hosts in your cluster licensed PLUS a licensed vCenter Server.