You can run the dedicated server and play a client at the same time, on the same machine.
The dedi server runs all the time, independent of whether the client is running or not.
@Jumper
You probably have a port forwarding problem, or a firewall problem.
Did you open the server ports in your router?
Post your port settings and maybe someone here can figure out your problem.
Also, you don't need another account or license to run the dedicated server, it is free.