Let's step back a moment. Why remove the engine to check for head gasket problems? Does the car burn coolant or is the oil contaminated with coolant? If not, you probably do not have a head gasket problem. Obviously you want to cure the cooling problem to avoid having to replace the head gaskets. The first step assuming you changed the coolant is to replace the thermostat and check the engine coolant temperature sensor. The ECT sensor controls the fans so maybe they are not kicking on when they should because of a bad sensor. If original then it could also be the water pump. Presumably the timing belt was changed at 80-100K miles, and it's very possible the pump wasn't replaced. Weeping from the pump because of a bad bearing is one common failure. But corrosion of the impeller itself can impede its ability to pump efficiently, and a slipping or worn impeller will do the same. Both are not unknown. It's also not unusual for the shaft to seize, but when this is happening it takes a toll on the timing belt, which might not last too long under these conditions. Replacing the water pump means taking off the timing belt, but then, depending on your mileage, it might be an opportune time to replace it as well.
By the way, even if you need to replace the head gaskets, you DO NOT need to remove the engine! Good luck!