A topic from AEG's L5R CCG Rules forum, copied for Oracle of the Void from a copy of the forum saved in October 2015, so pages it didn't save are missing. Players' names are left out; the rules officials are named. Rulings are as they stood on these dates; later rulings, the rulebooks and the Rules Archive may differ.
A player, 30 November 2007
If I discard ring of the void to draw a card, and then recover it with Togashi's Shrine, can I then discard the ring of the void again to draw another card?
Togashi's Shrine Limited: Bow Togashi's Shrine and discard a card: Search your deck and discard pile for a Ring. Show it. Put it in your hand.
Ring of the Void Open: Bow this card or discard it from your hand: If any player has more cards in his hand than you do, or if you bowed this card, draw a card.
Also does Togashi's shrine follow the wording rule of I have to search my deck first, then my discard pile?
Martin Lund (rules official), 13 December 2007
“Memory” of the status of individual cards ends when the card enters a deck or a hand, or otherwise becomes both face-down and out of play (such as after a card in the deck or hand is shown). This includes dishonorable status, ongoing effects, and whether or not an ability has been used.
See Comprehensive Rules: Effects - Tracking Rule.
"whether or not an ability has been used," is one of those things that does not get "remembered" when a card enters a deck or a hand. With Rings it might seem silly (since you only have one Ring of the Void in your deck), but consider what happens with non-unique cards like, say, Hida Daizu start getting shuffled into decks.
So there's a blanket rule that applies to all cards - non-unique, unique, and Event / Ring cards alike. Onces it goes face-down and out of play you can pretty much forget about any old states of or effects on the card. The only exceptions to that are listed in the linked section (mostly hedge cases for effects that put something out-of-play and then give you further instructions about that thing).
As to the Search part, I don't see any indication on the most recent printing of Togashi's Shrine indicating that the deck and discard pile need to be searched in distinct sequence. Treat it as a single search of a scope spanning both areas. Since your Discard Pile is face-up public knowledge anyway nothing is stopping you from seeing its contents while you rummage around in the Fate Deck. You can choose a card from either area.
If you're worried that you're somehow forced to pick all the Rings out of your Fate Deck before you can get them out of your Discard Pile you can relax. You can get that Discarded Ring of the Void back even if there's still a Ring of Fire in your Fate Deck.