Overrides on switched sides

I have a question to ask regarding overrides. I have recently switched sides from Saradomin, having obtained all three inventory peices for the Sara overrides (orb, hammer head, bowstaff) and turning one (the bow) into an override.
However, now in search of peices of Zamorak Armour, I have switch sides. I currently hold the Saradomin Hammerhead and Orb in my inventory, and having gained nearly 1300 tears from killing in a day, I have not obtained one override.

I was wondering why this was so – am I just really unlucky, or is it because I hold saradomin overrides – that I don’t seem to get any Zamorak overrides? Is this a glitch or an intended game effect? I would also like to know if anyone has held both Zamorak and Saradomin versions of an override in their inventory at the same time (i.e both Zamorak and Saradomin greathammer heads). That would clear a lot up for me.
If this is a glitch, it would be nice to have it fixed, but I don’t think it is. A few answers to my questions would be very much appreciaited.