I put shutoff valves on both hoses going to the heater, but really hot air is still coming out of the vents when driving down the road. Fan is off, A/C set to off.

Duct tape, it's the only answer.

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Corvettes and women are the same...You want to make love to them but they keep breaking down.

Oh common, Ken...it ain't HOT here yet!!

What is prolly happening is air from under the hood is getting into the cowl vents, and blowing into the cabin.
Since you have A/C, it is probably even worse, since you have a large opening on the pass side kick panel, for fresh air, that should have a close-off door. The seal on the door is most likely gone, or deteriorated to the point of not sealing at all, and this allows the hot air to invade the cabin.
Is the A/C hooked up, and working? Even if it doesn't work, you can put the thumbwheel(or slide lever, depending on what year model it is) to "Max A/C", and that will close the fresh-air door, and at least slow down the incoming heat wave.
Another very important thing to check is the rear seal that fits across the rear edge of the hood...if it is missing/damaged, there's nothing to stop the super-heated air from the engine from getting into the cowl vents, and therefore into the cabin.
Also, check to see how many holes in the firewall are not sealed...you'd be surprised at how much air can come thru a seemingly small hole around the speed-o cable, or even the accelerator cable.
hth


btw..what year model car is it?

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Joel Adams
C3VR Lifetime Member #56    
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"Money can't buy happiness -- but somehow it's more comforting to cry in a CORVETTE than in a Kia"

Sorry forgot to mention its a 76.