Chester-born former Everton trainee Waring, 21, is yet to make a first-team appearance for Stoke, but scored six times in 19 League One appearances on loan at Barnsley two seasons ago.
He then had 14 games on loan at League Two side Oxford United last season.
El Ouriachi, 20, joined Stoke from Barcelona after rejecting a new deal.
Their arrival adds to the nine pre-season signings made by Town boss Micky Mellon, on top of which he added young striker Ivan Toney on a half-season loan from Newcastle United.
Shrewsbury have started the new League One season indifferently, picking up just one point from their first two games, although they did knock Championship side Huddersfield Town out of the EFL Cup.
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Shrewsbury Town have signed winger Moha El Ouriachi and striker George Waring from Premier League side Stoke City, both on a six-month loan.