Panzer, I have never worked on a 450 before but I believe it works the same as my 500, both are 117. engines. First, the cold start valve only comes into play WHILE the starter is operating. The thermo switch is used to limit the amount of time the valve can stay on while the starter is engaged, and if it is used at all. Once the engine is wormed up, it is not used even when starting.
My first thought for your problem would be the control pressure regulator. A good check of the control pressure from a cold start to warmed up would be my next step.
I'm sorry, I am not good at explaining things, but I try.

Paul

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84 500 SEL (307,xxx miles)