I stumbled across this one and recognized a sad but common occurence. The caterpillar may have been in the process of shedding at the time (doesn't look like it to me), but regardless of this, it is not normal at any stage to be hanging limp like this one is. This is a sign that it was infested with a disease (or a parasite may have recently emerged from it), and it was basically "melting" internally. These limp caterpillars will sometimes dry on the plant, but usually fall to the ground. Sometimes another critter will come along and eat the evidence. If another of the same comes too close, it may catch the same disease and die as well.

This is indeed a Dill Worm or Black Swallowtail. When these pupate, they line up with their head upward on a silk pad spun onto the surface of a stick or nearly vertical surface. They also spin a silk "girdle" (a belt around the middle), then they shed and wiggle out of their skin. They end up head upward, suspended by the cremaster (a structure with hooks at the tip of the abdomen) hooked into the silk, and by the girdle around the middle. Usually they wander away from the food plant right before the pupate, and find a hidden or cryptic place. That is why you most often won't find pupae on the plant and why they seem to disappear. You are more likely to find them in a crevice in bark, on a wall, or hiding under some overhang.

But you have solved the mystery for me and I have learned something new. Thank-you.