How to troubleshoot a leak in an Aprilaire humidifier?
I can't quite tell whether you're seeing normal operation and mistaking it as a problem or if there's actually a problem. The white crumbly stuff is scale, also called mineral deposits. It is calcium and other minerals that are dissolved in drinking water.
I've drawn a line on the photo below to illustrate the path water should travel. It flows from the valve into the distribution tray at the top, trickles down through the evaporator pad, then collects in the tray at the bottom and exits through the drain hose. It's normal that the water that enters the humidifier will not all be evaporated - in other words, that some water will drain off when the humidifier operates. The amount of drainage will depend on how much the humidifier operates. If the indoor humidity is already close to the humidistat setting then the humidifier won't operate much and there will be little drainage.

The drain port at the bottom of the humidifier pad assembly should be inserted into the fitting at the bottom of the housing so that excess water goes through the drain hose. The humidifier pad assembly should sit fairly close to level and plumb in the housing so that the collection tray won't overflow. The water supply should be adjusted high enough that there's some drainage coming off the humidifier, but low enough to avoid overflowing the collection tray.
Based on the update of the question it sounds like water, as it trickles down the panel, jumps the gap to the plastic frame. A drip then forms and runs down the exterior of the frame, missing the intended drainage path.
The shim is a good idea. It seems reasonable to leave that in place and call it a fix. An alternative might be to modify the water panel. Maybe by crushing it down a bit on or near the edge you can make the gap from panel to frame large enough that rivulets of water flowing down the panel can't jump the gap.