If the assignment to the Director of Technology is simply for the approval, then you do not actually need that step.
1) What you should do is set up your request type as you have.
2) The tech group that responds to that request type (at level 1) should be set to assign the ticket automatically to the lead tech.
3) It sounds like you have the approval process set up correctly.
4) When the Director approves the request, then the status can be changed and the ticket can be worked on by the lead tech.
If the Director has to do something within WHD to the ticket first, then I'd suggest a different workflow.
1) Set up the request type without approvals.
2) Set up a tech group with two levels. Director at level 1, network administrator at level 2.
3) Director deal with the ticket, and escalate to level 2 to assign to the network administrator.