Transmission cooler question

Got a question for the knowledgeable folks here.
On a 1997 Chevy K series pickup, the transmission has bot a cooler in the radiator, and an external cooler mounted in front of the radiator.
Is the cooler in the radiator shell immersed in the radiator coolant, or is it a separate air cooled only thing that is just mounted coaxially with the radiator?
If it's in the radiator coolant, then wouldn't the fluid always be at whatever temperature the coolant is, which seems to be 195, as that is the thermostat open temp?
I burned up one trans due to a clogged external cooler. Added another outboard Hayden, in addition to the factory one that I replaced, so I now have 3.
Coolant flows out of the trans, into the radiator cooler, out of the radiator and into the factory external, then into the Hayden and back to the trans. With a temp sensor mounted in the side of the pan, trans runs at 120-130 with no load. Today, pulling a 26' enclosed car hauler with a 6000 pound load inside (trailer weighs 2600 empty), total load just under 9k, the temp was averaging 140-145, with one brief jump to 165 going up a long 6% grade. Outside temp was 75'.
Are those temps OK? Do I need to worry about it running too cool, or is there such a thing as too cool for a transmission?
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The cooler in the radiator is immersed in coolant. You can see that it is by removing the radiator cap and looking inside.

The transmission fluid can be hotter then the engine coolant, that is the whole point of the cooler in the radiator.

I wouldn't stick to the 195* thing too hard, the engine coolant can run higher than the thermostat rating.

If they are connected in series you now have one more thing to clog... I don't understand your rational, if the transmission failed due to a clogged cooler, the addition of another in series won't prevent that from reoccurring, adding one in parallel might.

Yes.
In extreme cold, there is.
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