I have this bit of code that's saving the position of the mouse cursor, but it's causing my bitmaps to flicker.

    ALLEGRO_MOUSE_STATE mouseState;      
    
    mouseX = al_get_mouse_state_axis(&mouseState, 0);
    mouseY = al_get_mouse_state_axis(&mouseState, 1);

I've also tried using events to save the position of the cursor, but that still causes flickering.

That means it is changing position rapidly. That means the values you're reading are changing rapidly. Did you get the mouse state before testing the values?