Same thing applies with a clutch fan as it does an electric fan really, a colder thermostat will start circulation through the radiator earlier but if the fan isn't engaged you're not really reducing the cooling system temp. With a clutch fan you unfortunately can't simply change the settings electronically, so with a lower temp thermostat you'd need to either find a clutch that engages at a lower temp use a fixed fan or convert to an electric fan/controller (my recommendation)
My question is how warm and what are your expectations? A early 3.8 obviously has the notorious head gasket issues so I can see your concern and desire to reduce temps but if the temps are in the normal operating range (190-210 depending on conditions) as measured by a real temperature gauge you're not likely going to preserve them with a simple lower temp thermostat. Before I'd throw parts at it I'd first make sure the AC condenser and radiator fins are clean and not blocked by years of grime and make sure the air dam under the bumper cover vents is still there intact, that's where the bulk of the air comes in to cool the radiator. It's actually a common retrofit on early MN12s to upgrade that air dam to one sourced from a 90s Grand Am which can be trimmed a little and put in place to grab more air.