Glad to see someone playing with a dipper, I have made a couple.
If your circuit works with a variable cap but not varicaps, its most likely an issue in the voltage control.
Firstly are you certain you havent got the diodes in backwards, what you say kinda points to that.
The you need to isolate the diodes from the osc dc wise with a blocking cap, then you need to isolate the diode with a high value resistor, 100k or higher, then the dc control wont damp the oscillation too much.
My latest dipper uses Led's as varicaps, and it has an Fm modulator to make it easy to find resonance, the original site has dissapeared unfortunately.