The current in the LM317 is about 10mA with no load. With no load the 10mA flows through R1 which causes the darlington to turn on (because 10mA x 1k= much more than its Vbe of 1.3V) when it MUST NOT turn on.
Therefore the value of R1 at 1000 ohms is much too high, it should be 10 ohms. If R1 is 10 ohms then it keeps the darlington turned off when the load current is zero or low and when the load current exceeds about 1.3V/10 ohms= 130mA. The LM317 passes up to 130mA.
Then the darlington begins to turn on and pass any load current that exceeds 130mA.
Q1 should not turn on if the load current is 120mA or less.
Since it turns on when you turn down the output voltage then I betcha that you connected the collector and emitter pins of the darlington transistor BACKWARDS, and the darlington transistor is a cheap fake one that is missing the diode from collector to emitter, or maybe it is not a darlington.
Oh, is D1 a zener diode instead of a normal diode?