Running those four LEDs in series at 350mA should be fine with that driver. You don’t need a current limiting resistor with a constant current type driver like the APC-12-350. As long as the LEDs are rated for 350mA or more, your are good to go. The only stipulation in this regard is that you will not be able to control the relative brightness of each LED individually, so you may get the wrong ratio of brightness for the various colors using this method.
Now about the failure. What happened? Did the power supply die, did one or more of the LEDs fail, or something else?
It’s pretty unlikely that the power supply failed, as it is current limited. More likely one or more of your LEDs died. Two things to consider there – first, LEDs are somewhat sensitive to ESD, so they could be damaged by a static discharge event. You really should take ESD precautions when handling them. However, more likely is that one or more of them overheated. You mention using a heatsink, but you did not mention using a thermal interface material (TIM) between the LEDs and the heat sinks. Without a decent TIM, the heat conduction between the LED and the heat sink is quite poor due to the microscopic irregularities of the two surfaces. The TIM fills those gaps and allows more effective thermal transfer.
Interface between LED star board and heat sink:
Here are a couple TIM’s cut in a star pattern specifically for LED star boards: