Would radiator trouble explain why my coolant tested positive for exhaust gases?
No, if it was showing exhaust in the coolant, there has to be a combustion-to-coolant mix happening somewhere. That can be a crack, a blown head gasket, or a combination of both. If the overheating situation was chronic, it's possible the gasket could have developed a leak after several incidents of too-hot operation. I'd strongly reccomend a good radiator and water jacket cleanout as part of your reassembly routine. One quick check for exhaust gas or compression pressure in the cooling system is to crank up the engine with the fan belt removed and water or coolant poured into the engine level with the base of the thermostat housing, then goose the throttle a few times before enough heat develops to boil the coolant. Any big bubbles or gushing of coolant indicates a compression pressure leak into the cooling system.
Jerry