Boncuk makes a good point. When you place a part of your body close to the board, you are adding a tiny bit of capacitance in parallel with some of the critical resonant circuitry on the VCO board. Imagine that your VCO is resonating with, say, 33.000 nH inductance and 29.0682 pF of capacitance (just a wild guess for example). Now, how much change in the capacitance would move the frequency by, say, 30 KHz (enough to move it out of the channel of your receiver)? Well, it would take 0.01073 pF. That's a very small capacitance change, and moving your hand nearby easily adds this much.
The way to fix this problem is to shield your circuitry. One of the easiest ways is to solder additional pieces of unetched copper pcb board into the shape of a box around your circuits. For the best performance, all DC going in and out (except GND) should go through a feedthrough capacitor, and RF should be taken out through a coaxial cable connector. Just as easy is to use sheet brass, commonly available at auto parts stores but they call it "shim stock". Buy it in a conveniently workable thickness, like .010 inches. Brass solders very easily and is easily cut with common tin snips. If you get the really thin stuff, like .002 inches, you can cut it with scissors too, but when it is this thin, it gets a bit wobbly and you run into another common VCO problem. Its called microphonics.
Microphonics is where your VCO is unintentionally modulated by nearby sound, or any vibration for that matter. I've made VCOs like this, where you just need to talk loudly at the VCO and you get FM from it! This is caused by very microscopic vibration of the components on the circuit board, or even the individual turns of your tank inductor. The solution is to use surface mount components and inductors that are potted in solid plastic.
Oh, and about that tuning wand. Well,that's a common problem too and not easily fixed. If your tuning wand is not metallic, then the problem is that your capacitor setting shifts very slightly when the pressure of the wand is removed. You could try a better quality variable capacitor, but they all do it to some degree. If your wand is metallic, then you are adding additional capacitance near the variable part, which is bad. Don't use a metallic wand. I've also seen where the part where the metal wand touches the capacitor is conductive and in this case that side of the capacitor should be the grounded side.