Nit: The transistor is a BC337; the number after the dash is the gain group which isn’t that relevant here. I tested one here.
What capacitor value and supply voltage are you using? Did you cut off the base pin? Which direction are you moving the potentiometer in when it stops oscillating – towards higher or lower frequencies?
The circuit works by charging up a capacitor until it reaches the voltage where the transistor opens. The 1k and the potentiometer controls the charge current, and the higher the combined resistance the longer it takes. It shouldn’t stop oscillating if it takes longer, though; the charge cycle looks exactly the same, just covering a shorter or longer period.
(The circuit may indeed fry the potentiometer if the transistor stays on; the 1k isn’t big enough to protect it if the wiper is too close to the connected end lug when that has happened. A potentiometer with a lower resistance can handle more current, though…)