It would pay to refer to your
previous thread describing the clock circuit.
The core of your problem seems not to be that the voltage varies when driving or not, but that your circuit requires fine-tuning to work well in the first place.
Your own observation that the base resistor needs to be between 20k & 33k tells me now that it will also be sensitive to temperature & voltage changes.
Now, what can be done to the circuit to make it inherently stable? When making amplifiers like that one common method is to add an emitter resistor.
Making it more or less like a constant-current driver it would negate any effects of temperature, transistor, & voltage variations.
What is the resistance of the coil? Did you add a collector resistance? Do you have room for one extra resistor & transistor?
