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Well here we go again, have 96 LT1/auto. No heat, fan runs, ac clutch works, It has climate control, so what allows the
the heat from the heater core to enter the cabin? Is it a cable with a trap door, or is it electrical, with a solonoid.
How did you determine that the heater core is hot? I would try to backflush it and see what comes out. Although, I am really leery of heater cores and hate to discover that I need a new one, I think flushing it in reverse is a good idea. If a bunch of crap comes out, you may have inexpensively solved your problem.
IF it is too far gone or leaks, well, find a list of handy expletives and see that you know how to use them.
By Climate control, I'll will assume you mean the C68 Automatic Climate control.
Air passing through the core is regulated by the Blend Door. The Blend door is moved by the blend door motor. On your 96 it is located on the bottom of the HVAC ducting under the passenger side of the dash. You have to remove the carpeted hush panel to gain access.
As a flat rate tech with similar customer complaints first thing I do is turn the fan on high and cycle the temp from hot to cold. As I'm doing this I listen for a chance in pitch from the air noise. If I can hear a pitch change I sell a heater core and collect my hour diag. If the heater core is plugged just replace it. If you flush it out it may work for a bit, but generally will just start leaking since got flushed all the crap out that was holding it together.