The spelling slough represents two meanings and two distinct pronunciations.
1. slough (rhymes with now) noun: soft, miry, muddy ground.
This is the kind of slough that John Bunyan describes in his allegory, The Pilgrim’s Progress:
Now I saw in my dream, that, just as they [Christian and Pliable] had ended this talk, they drew nigh to a very miry Slough that was in the midst of the plain; and they, being heedless, did both fall suddenly into the bog; the name of the Slough was Despond. Here, therefore, they wallowed for a time, being grievously bedaubed with the dirt; and Christian, because of the burden that was on his back, began to sink in the mire.
2. slough (rhymes with muff) noun: of a serpent or similar reptile, the cast-off skin. verb: to cast or shed the skin. Often used figuratively:
Putin, like Yeltsin, is constantly looking for ways to slough off responsibility for his decisions and their consequences…— Russia–Lost in Transition, by Liliia Fedorovna Shevtsova.
Slew, sometimes spelled slue, has more than one meaning.
The verb slew originated as a nautical term meaning “to turn a thing round upon its own axis, or without shifting it from its place. Slewed became nautical slang for “drunk” and a slew-foot was “a clumsy person who walks with feet turned out.” In Texas folklore, Pecos Bill marries a woman named “Slue-Foot Sue.”
The usual modern meaning of slew as a verb is “to turn a thing around on its own axis.” Here’s an example from fiction:
Near the top of the ramp a motorist in a gray Toyota panicked, slamming into the car behind it. Chrome and plastic hanging from its front, it slewed around blocking both lanes, effectively cutting off the Aviator. Robert Ludlum’s (TM) The Bourne Betrayal, Eric Van Lustbader.
As a noun, slew means “a very large number” or “a great amount.” For example:
Baltimore City legislators prepare for new Annapolis session with a slew of bills. City Paper, Baltimore.
A less common use of slew (also spelled slue) is in reference to “a marshy or reedy pool, pond, small lake, backwater, or inlet,” as in this description of a journey along the upper Mississippi River:
A continual variation of scene now opened to the view, marred only by an occasional ungraceful slew or marsh…
Some American speakers conflate the spelling and pronunciation of the words slough (miry ground) and slew (wetlands). They take their cue from Merriam-Webster whose entry for slough lumps the following definitions together:
1a. a place of deep mud or mire.
1b. a small marshy place.
1c. also slew or slue, a side channel or inlet
I’ll give the last word on the spelling and pronunciation of these words to The Chicago Manual of Style:
slew; slough; slue
Slew is an informal word equivalent to many or lots (you have a slew of cattle). It is sometimes misspelled slough (a legitimate noun meaning “a grimy swamp” and pronounced to rhyme with now) or slue (a legitimate verb meaning “to swing around”).
The phrase slough of despond (from Bunyan’s Pilgrim’s Progress ) means a state of depression. This is etymologically different from slough (/sləf/), meaning “to discard” (slough off dry skin).