Planting something so that the roots wll stabilize the slope and the topgrowth to hold back the mulch is the ideal solution. But we have no idea where you are, how large the area to be planted is, what the light aspect is and how steep the slope may be so any plant suggestions are up in the air at this point. And not a great time of year to be doing this in most cases anyway.

In the meantime, covering the newspapers and mulch with a geotextile matting or netting - something like this - should hold it all in place through the winter.

ETA: the jute netting is fully biodegradable so no need to remove before you plant in spring.

Choosing a highly textured mulch helps it stay put as well. I have used arborists’chips on a slope without a lot of movement. For plant suggestions your need to give the info GG 48 asked for.

Depending what was growing there already, IME newspaper and mulch alone may or may not kill everything, so if you have already done that, give it a few months of the growing season and then plant in fall so you aren’t cursed with an already weedy bed. If you haven’t already mulched and it is a sunny spot, covering it with black plastic for a season will kill everything IME and then can be mulched and planted.

GG and HNBabs have good suggestions. You might also consider digging a small trench at the base of your slope to catch mulch before it flows onto your sidewalk or wherever you don't want it to go.