Learn English – “Through the course” vs. “over the course”

I'd say "I am rapidly running out of time" - it's more accurate and descriptive IMHO since it expresses speed and motion, which is what the sentence is trying to convey.

Different dictionaries order their definitions in different ways. Some are diachronic, some order by senses, some by frequency of use - as with the Collins Cobuild.

The larger issue is that some words are used in specific situations, usual in certain contexts, which is more important than the order of their meanings as given in dictionaries.

For example: You can use the term "read" to mean "understand," as in, "I read you." This might be uncommon usage in an operating theater, but preferred in aeronautic communications.

Also: Word choice is often strongly influenced by surrounding words. More could be said on this subject if you provide the words and context in question.