Usually landscaping lights (incandescent and retrofit LED bulbs) work with AC or DC. If they were limited to DC, then polarity could matter (if the manufacturer didn't bother to put a 2 cent bridge rectifier in each lamp).

So either your replacement power unit is dead or your lamps require a certain polarity and the polarity is swapped. If you have a multimeter you can quickly check the output voltage, if you don't just swap + and - at the output and see what happens.

AC is not really any better for "range" (okay, if you're talking EHT lines running hundreds of miles/km then yes you can step it up/down easily), but AC transformers are very simple and almost never fail unless they are physically damaged or allowed to get wet. It's more likely the electromechanical timer has failed. Electronic power supplies have a limited life and will conk out eventually. They are light to ship and can be made fairly cheaply, despite their much greater complexity, at least they can be in Asia.