I rarely get an oily film on my soaping utensils since reducing SF to 0-3% in most of my recipes. When I was new to soapmaking, I used the default setting of 5%, but sometimes higher and my soaping vessels, utensils & so forth were always overly greasy. I hated washing and re-washing those things, which never seemed to de-grease. In fact, I had to toss some of them out because they began to smell like rancid oil after awhile! It didn't matter how much pre-wiping I did or how hot the water I used was or how much additional soap or detergent I used, it was a huge PITA doing the washing up. Alcohol, vinegar, Simple Green, bleach (not all together of course)... nothing helped as much as altering my recipes.
As far as handling the stick blender, I have a plastic cup that my soapy SB shaft sits in between uses during soaping. Once my soap is in the mold(s) & setting up, I wipe off the excess soap from the SB Shaft & run water into the cup with the shaft & sit in my sink so I won't forget to clean it asap. I also wipe all the utensils & bowls out, using either paper or old rags or the towels I cover my workspace with while soaping. The fabrics get set aside to dry out, then get washed when an appropriate load of laundry is next washed. No matter what, the SB shaft gets washed immediately because the lye can and does eat away at the gasket near the blade (yes, it happened to my first soaping stick blender). I may or may not wash the utensils immediately, depending on what's on my agenda. The bowls & other vessels (for multi-color soap mixing) get stacked together & placed in an out of the way safe area to await washing after any residue, of which there is usually very little, has a chance to saponify further.
All my dishes get air-dried. I rarely use a dishwasher, although I do have one and should use it more regularly so the gaskets don't go brittle, but I prefer hand-washing dishes. I'd really just as soon remove it from the kitchen & get the additional storage space.
When I make soap while traveling, however, which I often do, then everything gets washed up right away because I don't usually have a lot of spare stuff along for the ride, nor the space to keep soap-dirty dishes.