Actually the drift is thermal not due to voltage variations. Voltage
regulation is not an issue these days with the power line usually holding to
better than one percent.

In order to halt the drift, you need to replace some of the tuning
capacitance with negatifve temperature coefficient capacitors. How much you
use will depend on how much the drift is for a given temperature variation.
It drifts more at the high end because there is less capacitance involved,
causing a small variation to make the frequency move farther.

You can start with a negative temperature coefficient capacitor of, say 10
pF across the tuning capacitor and see if you can still align the local
oscillator. Mount the capacitor as close to the source of heat (the tube)
as possible and see what happens.

I have compensated many oscillators for thermal drift. You can also see if
perhaps the existing components are being heated by a power resistor and if
possible increase the distance. There is a bit of art involved but it's all
pretty basic.

The big problem, is where can you buy negative temperature coefficient
capacitors? They used to be plentiful but I don't know about these days.

Bob