The phrase "eyes shift" is correct and usable in written English.
You can use it to describe a change in eye direction or focus. For example, "Alice's eyes shifted away when her boss mentioned her tardiness."
I felt my eyes shift into high definition; my heart was racing.
His eyes shift upward to a circling vulture, a sentinel of inevitability.
His mouth drops opens like a child's, his eyes shift mischievously, his nose wrinkles.
When your eyes shift from one position to the other, they take time to move and land on the other location.
via GIPHY "If you ask about what a person heard, his eyes will shift toward his left ear to recollect the sound he heard, but if his eyes shift toward his right, he's about to fib," Bouton says.
The woman's eyes shift from side-to-side, and I continue.
Her eyes shifted over to me.
His eyes shifted nervously from Clea, settling on me.
His eyes shifted sideways, as if he were embarrassed.
Instead, the eyes shifted, there were tremors of disbelief, tinctured with moments of illumination as if this had to happen, and then the eyes shifted again.
Her eyes shifted back and forth, like those of a cat in search of something.
