"If This Is Winnetka, You Must Be Judy", a novelette by F. M. Busby; first published in the 1974 anthology Universe 5 (Terry Carr, ed.), which can be borrowed (free but registration required) from the Internet Archive. You may have read it in one of these compilations.
. . . his first memories might be flailing around like a baby in an adult body . . .
In his early time-years the skips were small, a day or two, and his young consciousness took them for bad dreams—to wake with unfamiliar sensations, body changed and everything out of size. Much later, waking in a hospital, he learned they were real.
"Do you use drugs, Mr. Garth?"
"No, I don't." A little grass now and then wasn't "drugs." "I'd like to know why I'm here."
"So would we. You were found lying helpless, unable to talk or coordinate your movements. Like a baby, Mr. Garth. Do you have any explanation, any pertinent medical history?"
So this is where I was, he thought. "No. I've been under a lot of pressure." That was probably safe to say, though he didn't know his body-age or circumstances. But in some thirty consciousness-years he'd learned to keep cover while he got his bearings in a new time. And eventually, as he hoped and expected, they told him most of what he needed to know about himself, and let him go. As sometimes happened, his research into the parameters of now was largely wasted; the time lasted only a dozen or so days. But the waste was not total, for when the following time came to him, he would still remember.
Once as a four-year-old he woke to middle age and panicked, screaming for his mother. He remembered being taken to the hospital that time, and did not look forward with pleasure to waking in it. But what had been would be. And he was certain there was at least one more infancy skip to be lived down someday.
He eventually meets someone else like himself, they realize it because they both know each other from a later time segment than the one they are in.
The black-haired girl walked by as he came out to the sidewalk, and before he could think, he called to her. "Elaine!"
She turned; frantically, he tried to think of a non-incriminating excuse. But her eyes went wide, and her arms; she ran to him and he could not resist her embrace. "Larry! Oh, Larry!"
"Uh—I guess I made a mistake," he said. His mind churned uselessly. "Perfectly natural. I guess I do look like a lot of other people."
She shook her head, scattering the tears that leaked onto her lashes. "No mistake, Larry." Her hands gripped his upper arms; he could feel the nails digging in. "Oh, think of it! You too, Larry! You too!"
His mind literally reeled; he felt dizzy. He breathed deeply, and again, and a third time. "Yes," he said. "Look, Elaine—let's go someplace quiet and have coffee or a drink or something. We've got to talk."
"Oh, yes! We have to talk—more than any other two people in the world."
They get together romantically thus changing previous 'fate', story ends.
"I've never tried to change anything before, Elaine. I guess I thought it couldn't be done. Or I was too busy keeping cover to think of making waves. I don't mean I followed any script; I didn't have one. But I went with how things were, and it all seemed to fit. Not now, though." He gripped her shoulder and turned her to face him. "I don't want you to die as you did."
[. . . .]
He had to say it fast. "I'm new here, Elaine. Straight from 1970. Nothing in between."
"Nothing? Oh, Larry, there's so much. And I've had only a little of it myself. Back and forth—and it's all so different."
"From . . . before, you mean?" His fingers ruffled her hair, then smoothed it.
"Yes." Her eyes widened. "Why, you don't know yet, do you? Of course not; you can't."
"Know what, Elaine?"
"How much have you had after 1970? How many years?"
"How much have I used up? I don't know—twelve years? Fifteen, maybe. Why?"
"Because it's not used up; it's all new!" Her hand gripped his wrist tightly, to the edge of pain. "Larry, I came here from '75—from a time I'd had before, married to Joe. But this time I was with you. This time we're together all the way."
