Personally I'd do what you're doing and just suck them up as much as you can. They could be like those tiny snails people get- they might be harmless to your fish, but if you don't want them, try sucking out as many as you can. I've never had any of those in my fish tank so I don't think they are normal beneficial things. The salt you are adding for your ich treatment might help to kill them as well. Most parasites don't like salt I believe(same as some fish, but fish tolerate it better). I'd try treating them with the ich medicine. The ich medicine might kill them as well. I think worms are invertebrates? Or at least would be sensitive to the ich medicine. I reccomend QuICK Cure. If you use that medicine I'd add maybe 40 drops to your oscar tank. Your oscars should be fine with a little bit extra of the medication, but it'll probably stress the worm things more and hopefully kill them. I suggest doing this anyways, or if they don't go away: take some in a baggy to a fish store (I don't reccomend a big chain fish store as the people there won't have any clue probably) or, take some pictures for us. I won't be able to help identify but there are many smart people here who might know what it is.
One more thing... about the overfeeding... your fish won't starve if you don't pour tons of food in.
Looked up on google about white worms in aquarium. Maybe it's this?