1997 Hyundai Elantra has heat but temp gauge does not rise.

Make sure your anti freeze in your radiator is up where it should be. Its possible you have an air lock in your system which prevents the fluid from circulating. To check on that when the engine is not real hot take your rad cap off and run the engine. and let any air that is in your system out. Make sure you have it cool enough you can run your engine and not boil out and burn you.Could be an air lock in your heater core as well to work out. Make sure you do not have a kinked hose. I even heard where some one installed his new thermostat upside down and then it would not work.

Changing gears and the temp gauge have nothing in common. First, check the coolant level. If gets low an air pocket in the engine will cause the gauge to go high, then back to normal. Next would be to check the wiring. could be the gauge/sensor wire is shorting to ground.

feel the rag on the passenger side jeep have bad heater cores

Wait till the engine is cold, remove the radiator cap, start engine and look for coolant gushing out of radiator or a lot of bubbles, its possible you have a blown head gasket.

sounds more like maybe bad temp sensor

Try replacing the radiator cap. A weak cap will allow coolant to leave the radiator and go thru the overflow tube to the reservoir.

Sounds like it's low on coolant. Check radiator. If low on coolant, heater may not get hot & temp gauge will read hot. The leak could possibly be a coolant leak..? Anti-freeze has a distinct odor when leaking from a hot engine. If that's the case, find the leak and have it repaired.

the reservoir level changes with the engine temp. the smell of anti-freeze would probably point to the heater core leaking.

you need to check your upper and lower rad hoses also your thermostat if in doubt take to a reputable Rad shop and have them look at it threy are usually very good with most rad and overheating problems

with the engine cold remove the radiator cap and check your antifreeze . im sure its low but now we have to find out why its low . look for leaks. filler up and get it warmed up and look around try to find your leak.'
good luck