Use AD5933 evaluation board for impedance calculation 1Khz-100kHz
You don't really have a very good situation for using the simple parallel plate capacitor formula and expecting accuracy. 

The equation ignores edge effects and you have slabs of acrylic or polycarbonate (I don't see any teflon there) with \$\epsilon_R\$ which could be as much as 4. There is also stray capacitance between your wires. And goodness knows what that work surface right by the plates is doing to your readings. A few pF is not a large capacitance. 

The formula is more accurate when the space between the plates is very small compared to the dimensions of the plates. 

You can put the copper plates in the middle of much bigger pieces with low dielectric constant such as FR-3703 PU foam, making the edges of the copper far from any other object. And keep the wire length down, keep them better separated and held in position. 

But you're likely still not going to be able to predict the capacitance to a high precision without using field solver software, and even then you would want to confirm with a precision measurement using good test equipment. The situation is too complex. It's possible to calculate capacitor values to high precision from first principles if they are made in an enclosed fashion, and standards agencies do that, but that is not your situation. 

Maybe if you look at what is actually going on (think about electric field lines between the plates, bulging out, taking paths through the plastic, table and between the wires etc.) you can model the situation as a fixed capacitance in parallel with a variable capacitance, and determine the ratio. Right now the setup is very messy.