Isn't it the tense? "They become" is happening now. I know that in English we do sometimes use the present to express things that happened in the past, but DL can't allow for that. The correct answers must be "They became very nervous" or "They have become very nervous". I assume the second sentence is the reason that "become" is in the dropdown - you can use it, but you have to add an auxiliary.
We should leave the meaning of GOT to situations where there is actual acquiring, obtaining, fetching, etc. Unfortunately get and got are often and frequently used/misused in American English. These words are more slang in everyday use and as you can see there are more precise words to convey those meanings. We have words which can give a better explaination of how we "get" something. In this case I would use the meaning of "became" (internal) rather than "got".