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Hi. If it only runs hotter going up a hill under a load its more likely to be a radiator flow problem. If it was a headgasket problem you could find your coolant low, if you ran the engine with the radiator cap off you may see a steady stream of bubbles, you may see oil in the coolant, or coolant in the oil. It sounds like you probably have a radiator problem. As the vehicle ages deposits form inside the radiator and block some of the passages. This in turn makes the radiator effectively smaller, and it can't keep up with cooling the engine when it makes more heat-under load.
A couple tests can be done. If you are worried about a headgasket leak there is a test called a block test that can be done. This tests the coolant for the presence of exhaust gases. The temperature of the radiator can also be checked with an infrared thermometer to see if all cores are flowing. If some are restricted some places in the radiator will be cooler because there is no hot coolant flowing through them.
Ok. In that case its a good idea to have the block test done on it. If it overheated bad there could be a problem there. If the coolant level doesn't go down, and it doesn't spit out of the bottle due to pressure, and the fluids are not mixing internally I doubt its the headgasket, but the test is simple and doesn't take long. You'd know for sure before going through the trouble of pulling the head off.