Hauling is a pretty low priority task in general. So if you really want to get hauling done, you're going to need to probably either disable a bunch of the tasks on a couple dwarfs, or make sure you only give a limited number of other jobs to be done at a time so that some dwarfs are doing those while others don't have anything else to do but haul.
Of course if your dwarfs are actually standing around doing nothing there's also the chance of having things accidentally disabled or screwed up in some way, like fumble-fingering a command and turning off hauling labors in your orders screen by mistake, or removing some stairs between your dwarfs and the food stockpile or something similar.
Edited to add: I don't generally play in evil environments, so is it also possible the dwarfs are just being horrified by their surroundings and not wanting to do anything? If that's the case, you might not have a lot done until/unless you manage to seclude them inside mostly, at which point they could still occasionally get horrified by reliving those moments but you might have more luck getting stuff done.
Edited again because I just noticed the "everyone in the military with half training all the time" thing. So do you mean that everyone is set to training mode and that you have orders edited so that only half are being forced to train at once? Or that you have the schedule set up so that a squad is training one month and off the next, with another squad "on" that month or what have you?
Either way, that might be interfering since I think military dwarfs will still occasionally train or do other things even if they don't "have" to by their schedule. I'm not sure though, since I generally only have a dedicated military, not "all dwarfs are military dwarfs." Anyhow it might not hurt to have a few dwarfs not in the military to take care of important tasks and/or making sure that some squads are specifically set to "inactive" at times. Again though, someone who does tend to run heavier/all military forts might be able to chime in a bit better here.