"walked carefully" is correct and usable in written English.
You can use it to describe the manner in which someone moved. For example: "He walked carefully through the dark alley, aware of his surroundings."
The man walked carefully up the hill.
The father walked carefully forward, holding the gun to his shoulder.
He walked carefully, quietly, his eyes blinking as they began to take in the dark.
Rinsing, I walked carefully across the wet granite floor to the tubs.
But by the end of the hour we have lived the woman's life and walked carefully with her into death.
One woman was bold enough to wear towering wedge heels as she walked, carefully, along the asphalt of Golf House Road near the 14th hole.
The exhibition was cordoned off and visitors had to walk carefully around the perimeter.
And young lads, all young lads, were walking carefully up to it, and touching it and dashing back.
With him you must walk carefully among the tombstones, reading the inscriptions, until you discover Benveniste's grave.
Most parts are mercifully well-grazed – but that means you have to walk carefully through herds of cattle.
I WALK carefully in the cool night air, focusing a kerosene lantern on the stone-paved path.
