It is grammatically correct, and I am sure that any native English speaker would understand, but you are correct in saying that it does not sound (quite) right. I think English speakers would say "We are in serious difficulty" or, even more likely, "We are having serious difficulties" Alternatively, if there were any doubt "Seriously, we are in difficulty".
"We are seriously in difficulty (or, trouble)." and "We are in serious difficulty (or, trouble)." have two different meanings. In the first, I am really serious about the fact that we are in trouble. In the second, I am stating a fact the the trouble we are in is, itself, very serious (has great consequences). Since this is lesson on adverbs, it would seem that only the first of the above translations makes sense.
Something is wrong with the way the sentence is being translated. The first time I translated "We are in serious difficulty"; I wrote: Siamo in gravemente difficolta" and was marked wrong and was advised that "Siamo in seria difficolta" was corrected. Now the same sentence is being translated as "Siamo in gravemente difficolta".
As a native American English speaker, I would never say we are in difficulty of any degree serious or otherwise.
You can be in trouble or in distress, but you experience, endure, know, or have difficulty. Those are a few examples that would be appropriate for the phrase serious difficulty.
Can anyone let us know if the Italian sentence is equally as poor a sentence as the English?