You probably are getting combustion on the chips without the chip insert/screen, even if they are soaked. I only use chips for fish, quick smokes or cold smokes. Try some chunks, and if they are kind of dry, use the "foil boat" method. I wrap all sides of my chunks tightly in foil except the side that faces up. When the cook is over, occasionally I will end up with a foil boat filled with white ash, but most of the time I end up with what looks like a piece of black charcoal. Folks here are pretty fussy about their wood. And for good reason. I don't have any of the fancy wood (have a couple bags of Weber chunks I'm trying to use up), so my wood is kind of dry. I haven't had any problems if I foil boat my crappy Weber wood, and the food tastes great. The smoke flavor is clean. But once my crappy wood is gone (or maybe sooner), I'm ordering some double filet chunks from smokinlicious.com. Seems pricey, but these smokers use such a small amount of wood, the per smoke price is very reasonable.
Since you said the taste is not dirty smoke or creosote, and more like a funky grease, I'm wondering if any of this will help, but it's a place to start. Is the grease taste at all like rancid grease? Like cooking oil that's gone bad? Or is it like a spoiled meat taste? Or more of a chemical grease smell like machine grease?