Try this: create a new folder for one NEF file from the D5600. Run Capture NX-D and navigate to that file; apply the Picture Control you’d like as the basis of your conversion – I prefer the Neutral one. NX-D might generate a couple of temporary icm profiles (some of them might give you the problems you describe), so at this point I’d re-start the program once again, open that one photo, go and copy the temporary icm file. It should be fine to work with in darktable.
Also, make sure you check the “Use this instead of…” setting in NX-D preferences:
