"I do not really like oranges."
Translation:Я не дуже люблю апельсини.
According to the dictionary, in its main meaning, помара́нча refers not to the usual oranges, but specifically to the bitter/Seville oranges, a different citrus fruit. The meaning 'orange' is marked 'colloquial' in the dictionary.
However, I believe this word is rarely used in modern Ukrainian. I personally have never heard the word помара́нча in colloquial speech. I'm pretty sure that in the shop you'll see апельси́ни and not помара́нчі.
By the way, Hrinchenko’s 1909 dictionary only has «помера́нець» (with the meaning 'orange') and doesn't even list «апельси́н». So it looks like your usage reflects the older meaning, while the modern usage in Ukraine is influenced by Russian.
I would translate it as "I really don not like oranges" (Я дуже не люблю апельсини). That is, one will not eat oranges if served.
"I do not really like oranges" (Я не дуже люблю апельсини), on the contrary, means one could eat oranges if served, but would not enjoy it very much.
Usually, you can use either genitive or accusative in negative sentences. However, this sentence is only negative in English, but not in in Ukrainian.
In English, 'don't' negates the 'really like', the main verbal phrase, so the sentence structure is like:
So, the English sentence tells us about the absence of real liking.
In Ukrainian, «не» only negates «дуже», and «не дуже» works as if it were a single adverbial phrase, meaning 'not really, moderately'. So, «Я не дуже люблю апельсини» is grammatically closer to 'I moderately like oranges', 'I like oranges not very much'. The structure is as follows:
So, the Ukrainian sentence tells us about a 'not real' liking. Of course, if you 'not really' like something, it mean you don't like it for real. So, those sentences are equivalent in meaning, but different in structure.
Since «люблю» is not negated, it's a positive sentence. In positive sentences, the direct object is normally used in the accusative case.
Sometimes you can use the genitive case for an object of the positive sentence, but then it means 'some amount of'. The English sentence is not saying 'I don't like some (amount of) oranges', it is about oranges in general, so this is why
However, there is a rub. Sometimes Ukrainian does allow objects in genitive without partitive meaning. For example, Iryna Bilyk sings «Я пишу́ тобі́ листа́» 'I'm writing a letter to you' with 'letter' in genitive, not «Я пишу́ тобі́ ли́ст». (Both sound natural.)
I don't know when such genitive objects are possible (I've tried to google some rules and found none), but for me, «я пишу тобі листа» sounds natural, and «я не дуже люблю апельсинів» doesn't. I'm not sure why. My guess: maybe it's because you can have 'some amount of oranges', but you can't have 'some amount of the letter' (so, the partitive/non-partitive distinction is neutralised here).
Anyway, I don't think «я не дуже люблю апельсинів» should be accepted.
P.S. I'm not sure whether adverbs should modify just the verb, or verb+object on the diagrams. I've marked them as modifying just verbs, but I'm not sure this is the correct representation.