The phrase makes sense because we commonly talk about our emotions, especially negative ones, by describing the effect of the emotion upon our digestive system. The phrase "drop with dread" for the stomach is one I have not heard before but its meaning is clear. The only word you need to understand is "dread". Whether the stomach drops, or freezes, or vomits hardly matters. Nobody would pay attention to the distinction. The only word that matters is "dread". A more common expression is "my stomach was tied in knots". There are also many colorful expressions that mention the digestive system in a similar manner, but which I cannot list here because they are too crude for this forum.

I must say I've never heard anyone use the phrase quite like that, with "with [x]" after it. For your stomach to drop is quite an unusual phrase. I understand it to mean that strange feeling in your body that can represent some deep fear. For me it's a similar feeling to the sensation of going downwards in a lift (which is perhaps where the phrase comes from). I wonder if anyone else understands it differently.