Hi,
You should be able to use that anyway. Probably with a capacitor and full wave rectifier. The cap will isolate the wall wart from the circuit for DC, but allow AC to activate the sense input. You'll need to use a non polarized cap maybe 0.1uf and then lower the value of the filter cap in the detector circuit to 0.1uf also. We'll have to see if this works ok.
Here's a method that should work regardless which lead the wall wart is common with on the output...
In place of the transformer in that previous schematic, use two capacitor 0.1uf each. They go from the AC wall wart output to a full wave bridge rectifier. The output of the rectifier goes in place of the transformer secondary.
So what we are doing here is detecting the AC by coupling the AC to the bridge, then use the bridge output as the DC detector source to drive the transistor. A 10uf cap should be ok for the filtering.