I've got a power supply adjusted to just over the forward voltage of the LED - about 3.7V.
Does that mean you're using a regular power supply with no current control?
High power LEDs (1W or more) normally use a special constant current (or controlled current) power supply.
If you don't know what the current is, you could be way-more or way-less than 1W. It will probably last forever at 1/10th of Watt...

LEDs (like all diodes) are highly non-linear and small change in voltage makes a HUGE change in current. On the other hand, with the proper constant-current power supply the voltage "magically" falls into-place.
With "regular little LEDs" we use a series resistor, and with a known-constant voltage drop across the resistor we can calculate/control the current. You
can use a resistor with high-power LED but it has to be a "high power" resistor (about the same wattage as the LED, depending on the voltage & current of course) and the resistor is wasting power & generating heat.
...Somewhat ironically, LEDs run a lot cooler than a regular light bulb of the same light output but an LED can't take anywhere the heat that a tungsten filament can take. So, it it's running cooler but it still needs a heatsink.