Let me weigh in again, ask a couple of questions, and suggest a possible answer.
1) What voltage is your power supply (and what is your power supply)?
2) what is the Vf of the 10W LED?
3) what current have you designed your constant current supply for?
4) does the LED current ever exceed (3) above?
My guesses to these are that (2) is quite close to (1) -- within 3V --, and that the answer to (4) is no. (I wasn't sure of this earlier, and I'm still not, but it explains what is happening)
The possible reason is that the Vf is large compared to your input voltage and the 338 can simply not turn on enough to provide the full current for the LED. As the LED warms up and it's Vf drops, the 338 is able to supply more current, leading to more heat, lower Vf, and what is normally called thermal runaway.
In your case thermal runaway is probably limited to (3).
You say that adding a heatsink causes the eventual final current to be lower. This is what I would expect if the thermal runaway theory was true.
If this is all true, a couple of things are required:
1) A higher input voltage (you probably need only a volt or so) OR a current regulator with a lower overhead OR a LED with a lower Vf
2) A heatsink on the LED
You did have some anomalous results, but your use of a 1/4W pot set to 0.1 ohms was about three different types of mistake, so I'll ignore what happened with this.