The first thing I would do in your case is to check the primary snubber for backwards components or open solder joints.  If you have effectively no snubber at the TinySwitch drain, the drain voltage can rise to levels sufficient to damage the TinySwitch internal mosfet. Also, make sure that the output rectifiers are installed in the correct direction. If they are installed backwards, there is no path for the primary magnetizing inductance to discharge into the load, and the primary switching mosfet can get damaged that way as well. Also, you should have some sort of load on the output as you bring up the supply. If you try to turn it on without a load, the output voltage can overshoot and the reflected voltage on the primary side of the transformer may be sufficient to damage the TinySwitch.

Standard operating procedure for bringing up a power supply of this type is to monitor the TinySwitch drain voltage with an oscilloscope using a 100X probe. As you carefully inch up the input AC voltage, the drain voltage should also rise. If not, you likely have an open component somewhere in your AC feed. If you have a digital oscilloscope, you can set the trigger to capture a single event, which would allow you to capture possible drain overvoltage events that can damage the device.

You will likely need some sort of preload on both outputs to improve secondary cross-regulation. I would suggest using around 10% load as a starting point.