This sounds like the electric radiator fan is not functioning. It is supposed to come on automatically when the coolant temperature rises above about 2/3 on the gauge. The electric radiator fan is activated by a radiator fan switch and a radiator fan relay. If the fan does not run when the motor gets hot; one or both of those parts may be bad. The fan switch is located on the thermostat housing. The fan relay is probably in a fuse block. There are also two other temperature senders on this motor in other locations; so be sure it is the fan switch you are changing.
You can confirm or disprove the blown head gasket theory by attaching a cooling system pressure tester to the radiator when the motor is cold, and not pumping any pressure into the system. Instead, start the motor and let it idle for three minutes, while watching the gauge. If the head gasket is bad; pressure will begin building up in the radiator to at least 5-10 psi within the first minute after a cold start. A normal engine will not build up pressure until it has run long enough for the thermostat to open (about 5 minutes or more).
If you add pure, undiluted coolant to the radiator; it will make the motor overheat. Coolant is designed to be mixed with equal quantities of distilled water in order to transfer heat properly.
Try testing the radiator cap on the cooling system pressure tester. If it takes 1/2 turn to lock in position and holds pressure; check the radiator filler neck to see if one of the stop tabs is bent or broken off. That would cause the cap to need a full turn to lock. A missing stop tab would also prevent the cap from sealing on the radiator. If this was the case; the filler neck or the radiator would have to be replaced.
What you wrote does not sound like there is a bad thermostat. A bad thermostat would make the temperature rise faster during acceleration.
If the water pump was bad; it would leak when the radiator was pressurized.
Once you get the radiator cap to seal properly, and get the fan to work, and are using coolant mixed with water; the overheating and loss of coolant should stop. If it continues; there is either an external or internal coolant leak. Pressurize the system again, and let it stay pressurized for 3 to 5 minutes; while you carefully check the water pump and the full length of all the heater and radiator hoses for leaks. If no leaks are visible and the pressure holds for that long; the radiator may not be cooling properly, the thermostat may be improperly installed or the thermostat may indeed be bad.