We just did Karazhan yesterday with my guild“s third raid with no priest at all (9th week of this raid). Me (Pally) and a shaman were healing, and a feral druid for those few times were we think we need three (Aran, Prince, Illhoof). Cleared all (except animal boss) in ~ 5 hours. Equip level is Karazhan or less - group setup is (sorted by equip level): Warrior Tank, Warlock, Rogue, 2 Feral Druids, Shaman, 2 Mages, Pally, Hunter (Twink).
Actually we find eg Moroes easier without Shackle than with a bad one. Decursing Magic debuffs is a pain for me for sure, but works well enough. Aran was a pain (4 tries), but everything else went quite smooth.
We've done Karazhan with as many as 3 Priests (2 Shadow), but as far as we have tried it doesn't really matter that much.. Added damage can outweigh a lack of healing easily.
We are not an "endgame raiding guild", my guild has done no more than Gruul and little poke at Void Reaver.