The problem had nothing to do with using the wall wart directly. The problem was you exceeded the max current and/or heat dissipation capacity of the smaller transistor which caused it to short circuit from emitter to collector long enough to explode. The wall wart doesn't 'push current' it's the load that determines the amount of current to flow and no more current flowed then the resistance of the relay coil allowed to flow. This can happen even with transistors 'rated' to flow a certain amount of current if you don't have enough heatsinking on the transistor that allows that 'rated' current to flow. A very important specification for a switching transistor is a graph normally shown in it's datasheet showing its SOA, safe operating area, which at all times must be within regardless of the actual voltage and current values being switched. This is a specification that considers the operating temperatures you are subjecting the device to.
Lefty