The newt story is "Heartburn" by Hortense Calisher. You can read it at UNZ.org. Here is the beginning of the story:
The light, gritty wind of a spring morning blew in on the doctor's shining, cleared desk, and on the tall buttonhook of a man who leaned agitatedly toward him.
"I have some kind of small animal lodged in my chest," said the man. He coughed, a slight, hollow apologia to his ailment, and sank back in his chair.
"Animal?" said the doctor, after a pause which had the unfortunate quality of comment. His voice, however, was practiced, deft, colored only with the careful suspension of judgment.
"Probably a form of newt or toad," answered the man, speaking with clipped distaste, as if he would disassociate himself from the idea as far as possible. "Of course, you don't believe me."
The man tells the doctor how he got the newt from a schoolboy:
"'I'll never get rid of it now!' he wailed. From then on it wasn't hard to get the whole maudlin story. It seems that shortly after Hallowell's arrival at school he acquired a reputation for unusual proficiency with animals and with out-of-the-way lore which would impress the ingenuous. He circulated the rumor that he could swallow small animals and regurgitate them at will. No one actually saw him swallow anything, but it seems that in some mumbo-jumbo with another boy who had shown cynicism about the whole thing, it was claimed that Hallowell had, well, divested himself of something, and passed it on to the other boy, with the statement that the latter would only be able to get rid of his cargo when he it turn found a boy who would disbelieve him."
Near the end of the story, the newt changes hosts again:
"O.K., O.K. . . !" he shouted suddenly, slapping his hand down on the desk and thrusting his chin forward. "Have it your way then! I don't believe you!"
Rigid, the man looked back at him cataleptically, seeming, for a moment, all eye. Then, his mouth stretching in that medieval grimace, risorial and equivocal, whose mask appears sometimes on one side of the stage, sometimes on the other, he fell forward on the desk, with a long, mewing sigh.
Before the doctor could reach him, he had raised himself on his arms and their foreheads touched. They recoiled, staring downward. Between them on the desk, as if one of its mahogany shadows had become animate, something seemed to move–small, seal-colored, and ambiguous. For a moment it filmed back and forth, arching in a crude, primordial inquiry; then homing straight for the doctor, whose jaw hung down in a rictus of shock, it disappeared from view.
I don't know the other story. Just in case the two stories really were in the same anthology, here are some of the anthologies in which "Heartburn" has appeared:
- Timeless Stories for Today and Tomorrow, edited by Ray Bradbury
- Best Horror Stories, edited by John Keir Cross
- Suddenly, edited by Marvin Allen Karp and Irving Settel
- The Cold Embrace, edited by Alex Hamilton
- Fantasy: Shapes of Things Unknown, edited by Edmund J. Farrell, Thomas E. Gage, John Pfordresher, and Raymond J. Rodrigues
- Nine Strange Stories, edited by Betty M. Owen
- 65 Great Tales of Horror, edited by Mary Danby
- Haunting Women, edited by Alan Ryan
- Night Shadows: Twentieth-Century Stories of the Uncanny, edited by Joan Kessler