Originally published in the March 1940 issue of Unknown, which can be read online at UNZ.org. You probably read this in his short story collection Starshine that was first published in 1977.
The story is indeed humourous in tone, while starting gruesomely:
I checked carefully. My feet were gone, so I wouldn't have to worry about them until the morning. My right hand, too: that was good. It would be awful to shake hands with Myra and have her find herself clinging to a disembodied hand. I pulled at the left. It seemed a little loose, but I didn't want to force it. This wasn't a painful disease as long as you let it have its own way. My face would come off any minute now.
There is a scene where he's running around trying to pull off loose skin and hide the sheddings so he looks normal to meet his love interest.
I am not generally a fussy man. A bit of litter around my two-and-a-half-room dugout on the West Side seldom bothers me. What trash that isn't big enough to be pushed out in the hallway can be kicked around until it gets lost. But today was different. Myra was coming, and I couldn't have Myra see the place this way.
Not that she cared particularly. She knew me well enough by this time not to mind. But the particular kind of litter might be a bit—disturbing.
I got the index finger off the piano and threw it and the left foot away, too. I wondered if I should get rid of the torso hanging in the hall closet, but decided against it. That was a fine piece. I might be able to make something good out of it; a suitcase, perhaps, or a rainproof sports jacket.
When she finally knows, she chooses to become infected like him.
Not quite. She was infected first.
She clung close to me and cried pitifully, "David, what are we going to do?"
I held her tight and didn't know what to say. She began talking brokenly: "Did it bite you, too, David? It bit m-me, the little beast... The Indians worship it. Th-they say its bite will ch-change you in to a snake... I was afraid... Next morning I began shedding my skin every twenty-four hours—and I have ever since."
The sheddings become used for artistic/craftsy purposes.
Amazing stuff, this cast-off skin. Regularly as clockwork, every twenty-our hours, the epidermis would toughen, loosen and slip off. It was astonishingly cohesive.
The nails would come off too, but only the topmost layer of cells. Treated with tannic acid and afterward with wool oil, it was strong, translucent and soft. It took shellac nicely, and a finish of Vandyke-brown oil paint mixed with bronze powder gave it a beautiful old-gold effect. I didn't know whether I had an affliction or a commodity.
Later there is a scene where he's accused of murder, because some leathercraft of his was made from human skin, he demands a DNA test to prove it is his own shed-skin.
Now, there is only one animal stupid enough to bang on a door when there is a bell to ring, and that is a policeman.
"So where'dja get th' ror material? Pleece analysis says it's human skin. What do you say?"
I exchanged a glance with Myra. "It is," I said.
It was evidently not the answer Brett expected. "Ha!" he said triumphantly. "Where'd you get it, then?"
"Tell you what I'll do, Brett," I said. I got a sheet of paper, poured some ink onto a blotter, and used it as a stamp pad. I carefully put each fingertip in the ink and pressed it to the paper. "Take that down to headquarters and give it to your suspicious savants. Tell them to compare these prints with those from the ornaments. [...] "
And closer to the end, he makes a living using a limited scope variation of the disease but hiding it as a normal beauty treatment - an injection to make the face shed (into a clay mask, so they don't notice), which rejuvenates the skin, then sneaking the shed-skin to his wife, who makes novelty masks for the same client without their knowing why it's so lifelike.
In the windup, I had it. An injection to cause the trouble, a lotion to cure or isolate it.
It was not an affliction, then: it was a commodity. The business spread astonishingly. We didn't let it get too big; but what with a little false front and a bit more bally-hoo, we are really going places. For instance, in Myra's exclusive beauty shop is a booth reserved for the wealthiest patrons. Myra will use creams and lotions galore on her customer by way of getting her into the mood; then, after isolating the skin on her face, will infect it with a small needle. In a few minutes the skin comes off; a mud pack hides it. The lady has a lovely smooth new face; Myra ships the old one out over to my place where my experts mount it. Then, through Myra's bally-hoo, the old lady generally will come around wanting a life mask. I give her a couple of appointments—they amount to séances—sling a lot of hocus-pocus, and in due time deliver the mask—life-size, neatly tinted. They never know, poor old dears, that they have contracted and been cured of the damnedest thing that ever skipped inclusion in "Materia Medica." It's a big business now; we're coining money.