In a perfect world, the seals are good, the adjustments are good, etc.
In the real world these things aren't perfect in a 20+ year old car and they leak a large amount of hot air and the shutoff valve helps by preventing the heater core from getting hot. If the heater core gets hot there is a good chance that it will warm the air conditioned air some.
The seals around the air doors within the airbox are made of foam or similar rubber material. As was already stated they might be deteriorated and not completely sealing the hot air from the mixing chamber. Another thing to check for would be the vacuum lines, if they are leaking the mixing valve door might not be pulling completly closed. The vacuum motor diaphrams also dryout and leak, so you really have several potential areas that could cause the mixing door to not function 100%. Installing a shutoff valve in the heater line will probably solve your "inside heat" problem, but it may also raise the operating temp of the engine slightly. You really need the full circulation of coolant in the entire system to manage those engine temps. If your engine is running on the cool side of what would be considered normal then you probably won't have a problem. If it's already running on the hot side of normal, then you might see it moving up into the area that would make you nervous on a long trip.
The heater core is really a small radiator that exchanges the thermal energy from the hot coolant and passes it into the heater airbox.... this provides a small measure of cooling for the engine.
If one has a digital readout for the coolant, you can actually see the cooling effects of the heater core while driving.
One other obvious area to check is the condition of the A/C refrigerant. Is it fully charged? If the A/C vent outlet temperatures have been slowly creeping up over a relatively short period of time (several weeks or months vs years) then you might need to have the system serviced if you haven't done so recently. Good Luck, and have a safe trip and enjoyable trip.