Translation:His mouth is full, but he is trying to speak.
FWIW, "his/her mouth is full" is the best rendering in English. "He has his mouth full" and "she has her mouth full" are also regularly used and are accepted. I see this is an example of a literal translation that doesn't work very well, and I don't think it would be used much (except when talking about teeth).
Funny, as I was doing this exercise, my father literally asked my mother if she had a full mouth (because she was eating and didn't reply).
I agree with you about the other constructions being closer to what I would normally use, but I had written in "has a full mouth" unconsciously and didn't find anything wrong with it so I came here.
I agree with the others in camp "he has a full mouth, but he is trying to speak". This should be an acceptable response. It comes down to personal preference. You could say either sentence and it would mean basically the same thing. Perhaps if it was in a book describing a dinner setting. The hypothetical story says "As John sits at the table with Cindy, he begins to get nervous. He has a full mouth, but is trying to speak, when suddenly the server drops a tray of food."
"His mouth is full, but he is trying to speak" for me, wouldn't quite sound right in the above example. It's not honestly that important, but they should both be accepted answers.
It is NEUTER (edited, as per comment below), but it exists only in the plural form ("pluralia tantum"). You can find the declension here: https://en.wiktionary.org/wiki/%C3%BAsta (click Declension if the table isn't visible). A brief description of singular-only and plural-only nouns is here: https://mluvtecesky.net/en/grammar/plural_singular.
In the context of food, "mouthful," is generally used with reference to what the "mouthful" consists of, so I'm not sure it works here. See the example sentences for definition 1 at https://en.oxforddictionaries.com/definition/mouthful, all of which refer to a "mouthful of X."
This reminds me of the mít hlad construction, which is not translated as "to have hunger," but rather as "to be hungry." Maybe this is similar, since the best option in English seems to be "X's mouth is full," rather than "X has a full mouth."