A cuteness overload is generally considered a good thing, whereas a cuteness overdose might be considered a bad thing (think about overdosing on sugar by eating too much candy, resulting in getting sick/vomiting).
While they do have particular differences, they can mean the same thing. "Overdose" is something people and animals can do, while "overload" applies more to technology or mechanics. In the phrase you give, there's no worthwhile difference, except how someone else might joke about what was just said.
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Has difficulty understanding even short answers in this language.
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