The problem is that your code doesn’t consider the fact that your character might (and likely will!) move past the target. You correctly calculate velocity as the product of character_speed and direction. Here, the character_speed determines the length of the velocity vector, so you need to cap it at precisely the point where your character would overshoot the target. It’s not a problem if it’s smaller (i.e. the character moves towards the target, but won’t reach it in the next frame), but it cannot be bigger! Calculate the distance to the target_position and divide it by delta to determine the maximum speed your character could move with the next frame without shooting past the target. Then simply take the minimum of your usual character_speed and that max_speed:
func _physics_process(delta: float) -> void:
var direction = global_position.direction_to(target_position)
var distance = global_position.distance_to(target_position)
var max_speed = (distance / delta)
velocity = min(character_speed, max_speed) * direction
move_and_slide()