Power Supply for xl6001 12V Switching LED Driver
Disable/Disconnect one of the LED drivers.Or disconnect the LEDs from one of the drivers.

I want to see if lowering the load on the LM317 helps.

UPDATE

Do you have a schematic?

I'd like to know exactly what series parallel means. I assume two strings of 7 White LEDs. 

Also how is the parallel was done?You should not drive two strings with one driver. If you do, you should balance the current to the strings.That would not be the cause here though.  Different matter all together.

Now the question is what is causing the failure?  The LM317 or Driver?

You said the output of the :LM317 (Vin to Driver) drops to 4.5v. If that is true then the problem must be the LM317 overheating. 

How long did you allow it to remain in failure mode?  Long enough for it to cool down and recover from the thermal shutdown?That may take as long as it did to overheat. 

FROM THE TI LM317 DATASHEET:
When an overload occurs the device shuts down Darlington NPN output stage or reduces the output current to prevent device damage. The device will automatically reset from the overload. The output may be reduced or alternate between on and off until the overload is removed.  

End of Update



Try Doubling the value of RS.

Very likely the XL6001 is going into thermal shutdown.  

The datasheet does not specify a max power, it just says "internally regulated".
and:


  
  Built in Thermal Shutdown Function
  Built in Current Limit Function
  


The datasheet does not specify its drive capabilities other than: 


  The  XL6001  could  directly  drive  5~8  Series  1W  LED units at
  VIN>12V.


That would indicate the driver can drive about 5-8 watts.  Not 14.



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When operating correctly, what is the measured LED voltage from the
LED driver output?
What is the max forward voltage for each LED from the datasheet?
Are the LEDs are connect in series?
Is the measured current through the LEDs 636mA?
XL6001 is configured as only a Boost Driver?