Man, the AutoZone site is FTL for me right. Way too slow.
But here is your wiring diagram.
CTS location: Under hood, center, upper engine area, rear of water outlet, mounted in intake manifold
No fan switch in sight.
Based on what you did, there are only so many things it could be. Grounding the CTS (don't do that again) made the fan come on, so to me that eliminates wiring harness issues and the relay. That leaves the CTS or the ECM. Is it throwing a CE light?
All according to this diagram: You have 2 fans, a main one turned on by the ECM, the other turned on via the AC being on or an auxiliary fan switch (where is this located?????). Each fan has its own relay. Does the one fan come on with the AC?
Change the CTS – they are $12. ECM is $100 (all at rockauto.com)