This idiom roughly translates as "the thing is, he/she is very busy", right?
However, a friend of mine on VK used "Хлопот полный рот", whereas on gramota.ru I saw "хлопот полон рот". Which one is correct, or are they both correct?
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"Полон" is the short form of "полный" and in this case they are absolutely interchangeable. However, "хлопот полон рот" is more euphonic and thus it is the preferable one.
Хлопот полон рот is idiomatic; хлопот полный рот is — well, like saying "shiver my timbers" instead of the more iconic "shiver me timbers". Полон is a bit of a relic here, from a time when short adjectives were more generally "indefinite", something like the Germanic weak adjective, instead of the more narrow predicative function ("is full") that they have today.