The paralysis is a big clue. Unfortunately, it fits a number of serious poultry diseases. It can be caused by starvation, heat and cold extremes, mold toxins, bacterial toxins in the soil, and an avian virus. It can also be caused by vitamin deficiencies. That's a broad set of possibilities to try to narrow down.
The bad odor to the poop is another clue that may point toward bacterial infection as well as mold toxic poisoning.
We wish to rule out the simple and common causes first. Mold. Is it possible the feed is contaminated by mold? You need to check. Feed has a very noxious odor when moldy. It all needs to be tossed out if you find mold. Treatment is to clean up all sources of mold and feed high protein and increase vitamins for a bit.
Since the poop is smellier than normal, it might be wise to go right to an oral antibiotic. I prefer amoxicillin which can often be found at feed stores under a "fish mox" label. The dose is 250mg per day for ten days. You will see improvement within 48 hours if bacterial infection is the culprit. Finish the whole ten days, though.
If no improvement, then it may be worthwhile trying vitamin E and B-complex for possible deficiencies that could be causing neurological symptoms. Get these at any outlet selling people vitamins. Pop a vitamin E and a B-complex pill directly into the beak daily for two to four weeks. If no improvement by the end of week two, it's probably not a vitamin deficiency. If there is improvement, a full four weeks or longer may be required for full recovery.
Last, if all of the above fail, then you may be looking at an avian virus. Sadly, there is no cure, and a necropsy after the chicken dies should be obtained to get a diagnosis as it would directly affect your whole flock.
