As I wrote in the 54600A thread the frequency measurement jumps, because that oscilloscope is not a counter and the frequency measurement there is just a tool showing aprox 4 digits frequency derived from the period measurement. You may get much more stable reading when you switch the Average ON (the 8/64/256 setting and you have to wait a little bit then), however.
The measurement of the frequency of a VFO (with the counter as well) has to be done either via a weak coupled inductance (a couple of turns) put 2-3cm off the inductor, or, in the case of a toroid inductor used in the VFO - coupled directly only at the VFO's buffered output (so after the second transistor - the buffer - coupled to the first VFO's oscillator transistor via a small, several pF large capacitor).
With coupling/loading the VFO with your probe directly on the LC or the transistor you will change the frequency of the oscillator (how much it depends on the actual frequency). While listening the beat in a radio you may hear the tone changes while messing around the VFO with the probe or your hands.
Building a "free running VFO" is a difficult exercise, such a VFO has to be properly shielded, mechanically perfectly stable (incl. the windings), its output buffered, and its LC has to be temperature compensated (even with 2-3MHz VFOs). Using the varicap diodes for tuning makes the situation even worse - either you TC the diode(s), or better use an air variable capacitor (mechanically stable one).
Better to go with crystal based PLL or DDS chips, like the popular Si5351, or AD9835/9851 etc for your aprox 2-3MHz VFO.