A stockpile will have empty bins/barrels in it, the default is equal to the size of the stockpile. Use <q> to change those. For a size 1 food stockpile, have 0 barrels reserved and a new, full(ish) barrel should take its place asap.
If it's an empty barrel, mark it for dumping <d,b,d, left click> and hope it gets replaced with something full.
Having a stockpile dedicated specifically for empty barrels (<q, s> a furniture stockpile) can help move those out of food stockpiles (unless they're being reserved
for that stockpile). Near your still and/or farmer's workshop is a good place.
How do I check what is inside a barrel?
k over it. It'll say the barrel type, and if there's something in it, it'll say that too. E.g. Oak rendered fat barrel or whatever.
Which is some info, but not all.
Use <k>, use +/- to highlight the desired barrel/bin (in case there's more than one item), and then hit <Enter> to actually see what's in it. That will show exact contents. Use +/- and <Enter> again to view an individual item, for instance if you want the exact contents of a prepared meal, or the value of something, or want to un/forbid/dump one particular item in a barrel.
Every time a dwarf starts the "give water" task, they interrupt it for Drink or Sleep etc, and then decide to do something else.
Sounds like your prisoner is too far from your meeting area, where dwarfs idle until they remember to do low priority jobs like this one.
\Back to putting some one on Healthcare only
I have Health Care on every dwarf all the time - even my distant miners, who have no other labor at all. If they're the ones who remember to save a dwarf from dying, it's all good.
I don't know if "Health Care only" is necessary (unless time is running out) - perhaps that and Refuse hauling, or food, or some other useful but sporadic job. HC is a low priority, unfortunately, but I think that's a "weighting" thing more than an absolute - that is, they're far
less likely to do it, but won't put other things ahead of it
every time. (That's the way it feels to me, anyway.)