Which sentence is correct? They went into 'a panic mode' or They went into panic mode?
In this sentence it is more natural without the article. Certainly if you leave off the word "mode" then the article is required, "They went into a panic", but "panic mode" is a noun that doesn't take the article unless you are wanting to imply there is more than one "panic mode" (perhaps "mild panic mode" and "ultra panic mode").
The indefinite article "a" is used when you are indicating one countable noun among many, and it doesn't matter which one, but if the noun is a singular concept, not countable, then no article is used.
Consider "He went into shock" and "he went into a state of shock": "shock" is a single concept, not countable, but there could be many "states of shock"