16 Volt on a 12 Volt LM317 Regulated Power Supply
Powering 96 led connected in parallel is a very bad design in the electrical perspective. Totally inefficient because it demands very high current. Try to group it in 12 parallel groups with 8 LEDs in series in each group. In this way you may power all these LEDs using 16 volt. Each series string will require 20mA and total current demand of the 12 groups will be 240mA at max.

Why 16v when the trafo is rated to only 12v? Because 12v is the RMS rate of the trafo ac output, but when it is rectified to DC and smoothed by a cap, the cap will be charged to the peak of the RMS output, which is 12x1.414=17 volt.

Why the 1.5A capable ic snubs at 560mA?
Because the ultimate current capacity is limited by the capability of the trafo. Your trafo must be rated to supply only 500mA at max.

The last thing is, this question must be posted in the electrical engineering site, not here.