I tend to think of "a schlepp" as a trek, an arduous trip, or a burdensome journey, and not something that is usually applied to simply going up or down a flight of stairs. Although, if tired, or in poor shape, one could certainly say, "I schlepped myself up the stairs," or, something like, "I had to schlepp four loads of laundry down to the basement," to emphasize the difficulty the task involved.
I want to describe someone moving in a slow and dissatisfied manner. This is the sentence I use it in: 'He schlepped down the steps of stone to the basement’s inky gloom, hearing Doreen shouting something indistinct.'
I don't think it's really appropriate for you to use "schlepp" to describe someone simply walking down some steps in a slow and dissatisfied manner--that doesn't capture what schlepping connotes. Schlepp pertains to the inconvenience, or difficulty, of doing something, much more than it does to dissatisfaction, and it's really unrelated to speed.
If you said things like, "To get away from hearing Doreen's shouting, he had to schlepp down to the basement's inky gloom," or, "To shut Doreen up, he schlepped down to the basement to get her a hammer," I think you'd be closer to using the verb schlepp in a more accurate way that reflects its connotative meaning.
Anyway, that's my take on it.