Sadly I have no more ideas as to what could be wrong, but I have had a look at the circuit and think that I understand how the oscillator works (maybe

).
IC1 and IC3 are a positive current generator.
Resistors R18, R19 and R20 set the current range.
VR1 adjusts the current.
IC2 and IC4 are a negative current generator.
Resistors R21, R22 and R23 set the current range.
VR1 adjusts the current.
The positive current generator charges one of the capacitors** C1, C2 C3 or C4 (depending on the frequency range).
The rise in voltage across the capacitor will be linear because it is a constant current charge.
Via TR5 and that cluster of transistors IC6 the voltage gets to IC7 which is a comparator.
When the voltage reaches some value, the comparator output changes state.
This switches* the current generator which is charging C1, C2 C3 or C4, to the negative current generator.
The voltage on the capacitor falls in a linear manner.
Again when the voltage at the comparator falls to some value, the comparator output will change state and we go back to putting positive current into C1, C2 C3 or C4 again.
So, at C1, C2 C3 or C4 we are generating a triangle wave.
The output of the comparator IC7 is a square wave.
And just how the sinewave is created, I dont know yet!
* The comparator switches the charging current via signal line "A" and the diode ring D1,D2,D3,D4.
** IC5 seems to do something strange on the low frequency range. It seems to act as a synthetic capacitor.
My brain now hurts, I must rest it!
JimB