It really does not want to provide enough current for the LED string, but does so at 11.5V, dropping as the regulator heats up.
That's because with a 19.5V input, you're dropping 8V through the regulator. That's 6.5V more overhead than the regulator needs. That's 6.5W of extra heat in the little lm317 package per amp of current.
Since your input voltage is fixed, you can drop some of that voltage with a power resistor on the LM317 input. Calculate the maximum current you want. Use that to figure the series power resistor value at 6.5V.
Per instance, say you want 2A max, your power resistor = 6.5V/2A = 3.25 ohms. At the maximum current draw of 2A, the input voltage of the regulator has now dropped to 13V, which is enough to get an 11.5V regulated output, still. The regulator now dissipates only 3W at this current, rather than 16W. The series power resistor dissipates 13W, taking that load off of the regulator.
At 3W, you will still probably need an efficient heatsink +- a fan.
At say 500mA, you could use a series resistor of 26 ohms. At 500mA, your regulator will max out at 750mW heat dissipation. The series resistor would dissipate 3.25W.
The series resistor caps the peak output current. Going above that chosen current value will cause the Vout to drop below your target. But it will increase the effective max
sustainable current. Obviously, this only works if your Vin is more than 1.5V greater than Vout.