Your oscillators are running on a too low frequency. They should run at about 285kHz. There are two resonant circuits on the var osc side, the parallel resonant circuit with the variable 100uH inductor and the 3.3nF capacitor and the series resonant circuit with the linearization coils (4 x 10mH) and the static antenna capacitance. The latter's resonant frequency has to be slightly lower as the one of the oscillator itself. There is a complex interaction between both which influences antenna sensitivity, pitch range and linearity.
Possible causes may be
-bad tuning of the variable inductors
-too much stray capacitance around the 4 linearization coils
-bad choice of the linearization coils with a too low SRF (one should use the recommended 3pi wound 10mH inductors by Bourns-Miller or their Coilcraft equivalents)
-bad positioning or wiring of the linearization coils (too much coupling between them)
-too much parasitic capacitance seen by the antenna.
The var osc without the antenna coils connected should run at about 275kHz. As soon as you connect the antenna circuit, its frequency should go up by ~10kHz. When touching and grasping the antenna now, the frequency should go down by 5kHz. If that works, you may tune the fixed pitch oscillator for zero beat with your hand at about 50cm/20" from it. Now you should get a pitch range of about 2.5 kHz and all is fine.