Funny, as I was thinking about your problem this morning, I thought I should have suggested looking for the wiring diagram on the relays, and testing them. (No, I haven't been obsessing over your Trans Am all this time, but you know how things just pop into your head?) Usually, but not always, the thermoswitch supplies the ground to the relay when the thermoswitch closes. If that is the case with your car, then you could put the relay in and ground the wire from the thermoswitch, and the fans should come on (if there is only one wire at the the thermoswitch it has to be a ground.) The relays can be tested off the car by feeding the coil from a 12 volt source, and measuring resistance (or using a powered continuity light, etc) across the output contacts. Ordinarily, you hear a click and the resistance changes from infinity to near zero.

The fans themselves can be tested by powering them from a 12 volt source. (They can draw quite a bit so be prepared for a couple sparks.) The relays can be tested the same way, as above. The thermoswitch can be tested in a pan of water with a multimeter (although there may be some now that are set slightly over 212 -- but I doubt it.) or you could infer that it is not working if all else is. So that covers the possibilities, other than the wires themselves, which you can check with an ohmmeter. (Also, you can verify voltage at the supply side of the relay (meaning the fuse is OK))

... OR ...

The other possibility, if everything seems to be working, is that the thermoswitch to relay connection is not direct. In an effort to save weight, parts count, etc. more and more computer modules are used for various functions. The fans could be turned on whenever the engine control module senses a particular temperature from the sender -- in which case a thermoswitch is not required. You'd need the car's wiring diagram to know this -- or call the dealer maybe... or ask at a parts store.

On most cars now, things that used to be really simple are fairly complex. Even interior lights are rarely turned on directly now, but are instead under control of the body computer. All these things are black boxes, from a diagnostic standpoint, but the actual electronics involved can be thousands of transistor junctions where there used to ba a wire and a switch. Personally, I'd rather have the interior lights in my Honda simply go off when I close the door, rather than having them slowly fade out after a delay -- but I'm an old fart.

Other thoughts:

With one of the relays removed, you could check the coil voltage at the engine temperature at which the fans should be running.

I wonder why there are separate relays for each fan? maybe they come on at different temperatures, in which case control by module would be likely.

Also make sure the thing you replaced was the actual thermoswitch, as opposed to the temperature sender (for the gauge).

As I'm writing this, I'm thinking you really need the wiring diagram for the car, if the obvious tests don't pinpoint the culprit.

If your are stuck and need to use the car, you can always replace a relay with a toggle switch wired to two short leads with terminals to match the relay. Turn the switch on when you are driving, off when you are not. (The fans are not shut off w/ the ignition if they share a fuse with the headlights.)

It sounds like you are pretty adept at this stuff -- but do beware of creating sparks around the battery (low probability of anything happening, but I keep remembering my wife trying to jump start our car: she wiggled the cable to improve the connection, and boom, the entire top of the battery blew off, followed by an acid shower.

Also, take all this advice with this caveat: I'm an old fart who used to twist wrenches professionally quite a while ago, but only work on my own cars now. My own cars have been very reliable, so I'm less in the loop than I used to be.

Watch the papers for reviews of my next novel.