In this example, "sense" is correct, so you could say: "nouns with both a countable and an uncountable sense". You have more than one "sense", but you identify each of them indivdually with the articles "a" and "an". You could also speak of an unspecified number of "senses" which are either countable or uncountable: "nouns with both countable and uncountable senses".
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