lets make a thread

[night ether]

private bool canAttack=true;

else if (attackDistance >= distance && canAttack)
        {
            RandomlyChooseAttack();
        }

void CoolDown()
    {
        timer -= Time.deltaTime;

        if (timer < 0 && cooling && attackMode)
        {
            cooling = false;
            canAttack = true;
            timer = intTimer;
        }
    }

void StopAttack()
    {
        cooling = false;
        attackMode = false;
        anim.SetBool("DownSlash", false);
        anim.SetBool("SideSlash", false);
        anim.SetBool("UpSlash", false);
        canAttack = true;
    }

that should do it

so the logic is
to start with you can attack
once an attack has started you cannot attack again until the cooldown period is over or the attack is stopped

[coarse thistle]

I tried it. RandomlyChooseAttack is being called constantly and timer is not counting down after these changes

In the previous version of the code RandomlyChooseAttack is being called only when enemy attacks

This script is attached to enemy. It's enemy AI for a 2D platformer kind of game

[night ether]

do you still have the canAttack=false in RandomlyChooseAttack you need this

should not have canAttack=true in TriggerCooling

take it out