2005 Toyota Camry Overheating

The part of your question is confusing me that your radiator is cool to the touch and you are adding fuild. There is only three places for the fuild to go (FLOOR, ENGINE, TRANSMISSION) no white smoke, did you look at the oil itself is it milky? Did you check your transmission fuild? If the passes were clogged you would build up pressure and heat.

The oil isn't milky and the transmission fluid is fine.

"Drinking coolant like crazy".....you might consider performing an engine compression test. Also...you might have AIR trapped in the cooling system.

Both hoses are hot

It appears you might have a restricted radiator. Consider this "old school" test. D/C the lower radiator hose and remove the radiator filler cap. Take a garden hose and place the open end of the hose onto the lower outlet of the radiator. Try to get a good seal between the hose and the rad. with your hand and a rag. Have some one turn the water on FULL BLAST. Water should flow up through the rad and come squirting out of the rad. fill hole about 12 inches. You have accomplished 3 things. Cold water should have closed the thermostat preventing most but not all water from going into the engine. You reverse flushed the radiator and the water pump. You tested for restriction in the radiator. Good Luck

BTW....you might still consider that COMPRESSION TEST. We have not ruled out a bad head gasket. Also when you refill these cooling systems today you should be using a special cooling system refill FUNNEL. Follow the directions carefully. This funnel helps bleed air out of the cooling system.

Turned out to be bad motors on my radiator fans. Replaces them and I'm as good as new.

Great. Thanks for the update.

I was losing rad fluid thru my cap because a small piece on the bottom of the cap broke off making the cap loose. Put on a new one, fixed the problem.

Hopefully this fixes my problem