the runaround
Idiom is correct.
Any idiom can be correct. The trick is to learn which one means what!
This is not an idiom. When you see AS ___ AS ___ you have A Simile. The correct simile is "on the tip of his tongue."
An idiom is a phrase that makes no sense unless you know the definition. Can a nose actually run somewhere? No, so this is an idiom.
If you are exhausted but keep going anyway, you are running on empty.
This is NOT an idiom -- when you hear AS __ AS __ you have A Simile. The correct simile would be "we're all in the same boat," meaning "we have the same circumstances for everyone."
The correct idiom is "a frog in my throat," meaning that your voice is hoarse and croaking.
The phrase "Always go in search for the correct idiom" is #39 on a list called "The Bestest Guide to Gooder English". The idiom in the sentence is "go in search for" and it's not incorrect,really, but just kind of strange. "Use appropriate idioms" or "If you're going to use an idiom, make sure it's appropriate" would be better, I imagine.The one on that list I haven't figured out is #40, "3D96#" Anybody know that one?
As an idiom yes.
"For free' is grammatically correct. It is an idiom of the English language.
This is not an idiom. It means exactly what it says in grammatically-correct fashion.
It's not an idiom because you can figure out the meaning if you think - if you're running around in circles, you're not getting anywhere. It's a metaphor for fruitless work.