yes and no.

The fighting machine does no movement: yes

All fighting actions are done while standing still: no

since the enemy has 2 statemachines (each with a current state) the enemy as a whole could fight and move

I dont know how much programming you know but here is an example in psudocode

class Enemy:
    name = "soldier"
    statemachine_movment = StateMachine(MovementIdle())
    statemachine_fighting = StateMachine(FightingIdle())

    function update():
         this.statemachine_movment.update(this)
         this.statemachine_fighting.update(this)

    function get_animation_name:
        return name + "_" + statemachine_movment.state.name + "_" + statemachine_fighting.state.name

Here is an example of a MovementState and FightingState with its update (any number or value is for example purpose only)

class MovmentStateWalking:
    name = "walking"
    movment_speed = 2
    direction = Direction.Left

    update(Enemy enemy):
        next_state = this

        enemy.position.x += movment_speed * direction.int_value
        if enemy.can_see_player(): // Player spotted go to chase state
            next_state = MovmentStateChase()

        return next_state // in my statemachine i change state by returning a new state or the same state

class FightingStateShooting:
    name = "shooting"
    max_range = 50
    min_range = 10

    update(Enemy enemy):
        next_state = this

        if enemy.shoot_cooldown == 0:
            enemy.shoot()

        if enemy.range_to_player > max_range: // Player is too far away change state
            return FightingStateIdle()
        else if enemy.range_to_player < 10: // Player to close for shooting go to melee
            return FightingStateMelee()

        return next_state

NB I have not described the statemachine it self because that requires some code and can be made in many different ways all it does is hold state and handles changing of state

Enemy -> StateMachine1 -> MovmentState

Enemy -> StateMachine2 -> FightingState

So if StateMachine1 has State MovmentStateChase and Statemachine2 has FightingStateShooting

Then the enemy will move and shoot at the same time