I struggled with a problem like this on a Glassair III some friends
built.

Had a transistor in an Darlington emitter follower configuration. The
dimmer pot
drove the base. The lights were from the emitter to ground.

Trouble was, the thing was oscillating when at mid brightness
positions. Too much
capacitance on the output. A known problem with emitter followers.
One forgets
that they still have gain at a few tens of mhz. When it took off you
could hear it
in several of the radios. Darlington configurations have worse
stability problems.

I solved it by puttting about 100 ohms in the base right at the
transistor.

These circuits are designed by people not very skilled in the art.
They also suffer
the problem that if a bulb burns out shorted or there's an inadvertent
short on the
string of lights, the transistor fails. There is nothing to limit the
current.
That will usually take the pot too, especially if
it's near the high end of its range. The 100 ohm resistor will solve
that, too.

If it's not a darlington, the resistor will have to be smaller.

The cool way around all this is to design it with a P-FET power device
configured like an op-amp.

Bill Hale