What you're seeing is the effect of poor transformer regulation. Small transformers have very poor voltage regulation - 30% or more is very common. Big toroids like you might see in an audio amp have great regulation - 5% or less with the >300VA units.
Look at some specs for AC/AC toroid power transformers. You'll see that they might spec 15VAC @ 1A with 20% regulation. It will actually output 18VAC with no load, and 25.5VDC minus 1.4V diode losses = 24.1V after rectification and filtering caps.
If you load it down so it is actually drawing its full rated current, it will drop down to its rated 15VAC and you'll get 21VDC minus 1.4V diode losses = 19.6VDC.
Your wall-wart, like all unregulated wall-warts, has a really poorly regulated AC/AC transformer inside which gets rectified and filtered. Because it's poorly regulated the no-load DC votlage is higher than it should be, like in the example above. It's common for 12VDC unregulated supplies to be 15-16VDC at no load.
It's also good to keep this in mind when designing power supplies. If you size your filter capacitors in a power supply so that they will handle the nominal output of the transformer after rectification, the actual (no-load) voltage after rectification and before regulation can exceed the voltage rating of your electrolytic caps.
For example, 15VAC rectified should be 21V minus 1.4V diode losses = 19.6V when rectified and filtered, and I might choose 25V electrolytics for filtering caps. Poor transformer regulation means that 19.6V could easily be 25V or more - and poof go the caps. 35V or 50V would be a better choice in this example.
But in your case, I would just find a LM7809 voltage regulator to knock it down to the 9V you need. There's no need for the slightly more complicated LM317 here if you're just replacing a battery.