a. I want to eat chocolate, I’m allergic.
Most people would fix this by adding a conjunctive adverb:
a. "I want to eat chocolate; however, I’m allergic."
Would it be incorrect to say "I want to eat chocolate; I’m allergic."?
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Gramatically you can use semicolon without conjunctive adverb after it. It is often the case when two independent clauses are close in meaning.
The sky is clear; the stars are twinkling.
The waves were crashing on the store; it was a lovely sight.
But semantically there may be differences in meaning that are conveyed with such adverbs in concrete examples of sentences, and without those adverbs you won't convey the desired thought. I think, adverbs should be used where there isn't such a closeness in meaning between two independent clauses.
I want to eat chocolate; I’m allergic.
This is not correct. It does not represent the meaning you want to convey. It is two related independent clauses which I agree upon, but the idea of you being allergic to 'chocolate' isn't represented properly here. In such cases I would prefer using sentence like
I want to eat chocolate; I am allergic though.
The above sentence represents your want for eating chocolate, but your restriction because of the allergy.
Or simply use
I want to eat chocolate, but I am allergic.
I want to eat chocolate; however, I am allergic. (your own sentence suggestion)