Does every psychiatric drug mimic a neurotransmitter?

Not exactly: Psychiatric drugs work in a number of ways. Most affect brain systems that increase or decrease a particular neuroregulator, e.g., blocking a receptor or slowing reuptake back into the releasing neuron. Others are mimics (agonists) that work by fooling the brain into thinking a system is more active than it is.

Psych MEds: Psych meds are designed to increase or decrease neurotransmitters depending on the condition, they may work or may not work its 50-50 usually.