It really depends on the device, and I don't know a lot about palm pilots. The 5.2V spec scares me though... it's too precise. Some wall wart adapters have voltage regulators that provide exact voltage: in my mind if you spec 5.2V you want a regulated voltage.
The Radio Shack multi-voltage ones (and most wall warts, really) are unregulated. The voltage rating is accurate but you have to understand what it is saying. If it says 6VDC 2000mA, then at 2A (2000 mA) it will be close to 6V. BUT!!! with no load, or 200mA, or anything below 2000mA, the voltage will be higher. At low load it will be a LOT higher, 6V units are often 12V at no load and I've seen higher.
A regulated 6V supply will put out 6V from no load up to full rated power.
I would open the original broken power supply. If all you see is a transformer and rectifer bridge then it's unregulated and the Radio Shack one will be fine. This is how theirs work, the transformer has a multi-tap secondary and the voltage selector slide switch just selects different voltage taps.
If you see a bunch of electronic parts in the original supply it's probably a regulated supply - I would avoid the cheapo unregulated supply. You could easily fry the palm pilot. AT the least, I would put a meter on the power supply output, select the lowest voltage, and plug in your pilot. Then turn the supply up one notch at a time until you get as close to 5.2V but not too much above it as you can. This could still be risky: the pilot supply may have some intelligent battery charging built in.
