The first story sounds like "Alas, All Thinking" (1935) by Harry Bates. The plot is as you describe. The story takes the form of a report, and the main character makes his announcement a few pages in:
He answered cryptically, bubblingly, enjoying our puzzlement with
every word.
"Because Humpty Dumpty had a great fall. Because thought is withering,
and sensation sweet. Because I've recovered my sense of humor. Because
'why' is a dangerous word, and makes people unhappy. Because I have
had a glimpse of the most horrible cerebral future. Yes!" He laughed,
paused for a moment, then said in a lower voice with dramatic
impressiveness, "Would you believe it? I have terminated the genus
Homo Sapiens."
The main character describes his first encounter with one of the adult future humans:
"I saw a man; or some kind of a man. He sat right in front of me, nude
from the waist up, and covered as the floor was covered from the waist
down. How shall I adequately describe him!
"He was in some ways like an unwrapped mummy, except that a fallen-in
mummy presents a fairly respectable appearance. And then he was
something like a spider--a spider with only three legs. And again,
looking quickly, he was all one gigantic head, or at least a great
mass on whose parchment surface appeared a little round two-holed
knoll where the nose customarily is, lidded caverns where the eyes
belong, small craters where the ears commonly are, and, on the
underside, a horrible, wrinkled half-inch slit, below which more
parchment backed almost horizontally to a three-inch striated and, in
places, bumpy pipe.
"By not the slightest movement of any kind did the monster show he
knew I was there. He sat on a high dais; his arms were only bones
converging downward; his body, only half the usual thickness, showed
every rib and even, I think, the front side of some of his vertebrae;
and his pipe of a neck, unable alone to support his head, gave most of
that job to two curved metal pieces that came out of the wall..."
The story is available to read online in the context of its original publication at the Internet Archive. An illustration on the page preceding the start of the story may be familiar.
ISFDB does not seem to have any entry for a collection containing both this story and Stephen King's "The Jaunt" (1981), which I agree is the second one that you describe above. You probably recall that story's memorable ending:
The thing that had been his son bounced and writhed on its Jaunt
couch, a twelve-year-old boy with a snow-white fall of hair and eyes
which were incredibly ancient, the corneas gone a sickly yellow. Here
was a creature older than time masquerading as a boy; and yet it
bounced and writhed with a kind of horrid, obscene glee, and at its
choked, lunatic cackles the Jaunt attendants drew back in terror...
"Longer than you think, Dad!" it cackled. "Longer than you think! Held
my breath when they gave me the gas! Wanted to see! I saw! I saw!
Longer than you think!"
