"I'm dreadful when it comes to making a mess!" ......?

It's correctly worded, but it's not a typical sentence construction, and the word dreadful can mean numerous different things (which are at least somewhat related in most cases). Inspiring fear, inspiring awe, and terrible / bad, in American, Australian, and British English. In British English, it also can mean something similar to 'overly dramatic'. I'm unaware of any Australian-specific additional meaning behind the word, but I imagine it's entirely possible. I'm also aware of any English dialects or subcultures within English speaking groups where the word takes on additional meaning, but again, I'm sure there could be.

In this case, the word could either be considered to mean 'terrible', thus implying the person is messy, or given the understood implied meaning of the sentence, could simultaneously that the person is so messy that their messes are so terrible that they inspire awe and/or fear.

I could definitely how you could see a double-meaning behind this; possibly meaning that the person is messy, or possibly meaning that they're not messy. I actually think that it might have an ambiguous meaning technically speaking, now that I think of it a bit more. I suppose that it might help to note that the word dreadful (and I just looked it up to see if this was even implied in the dictionary definition--it's not) in the meanings of inspiring awe or inspiring fear, generally always implies whatever is most unfavorable in situations where the meaning is ambiguous.

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Edit:

Yes, if a person said "I'm dreadful when it comes to being neat!", that would imply that the person is not neat at all.

To me it says "I'm awfully good at making a mess..."

That is to say, this is a messy person.