The two-sentence rewrite
Nexisa resolved not to give up until she had achieved her dreams. Her
determination was stronger than ever.
is very similar in both the meaning and the emphasis (other than focus) of the two statements. As regards style, I suspect different people might prefer one or the other; some might even say they'd make a choice based on context.
The point is that there are really two connected statements here, but not so intimately connected that they can't be put into consecutive sentences. So if one must allocate function here, surely the absolute phrase is modifying the whole [of the rest of the] sentence. And as you indicate, not just the noun or the verb phrase. I don't call 'sentence modifiers' 'adverbs/adverbials' as they modify other than just the verb phrase.
The situation is subtly different from that when adjectives obviously referring to the following noun are used in absolute constructions:
Exhausted after their climb, the boys sat on a convenient rocky ledge.
Here, the two-sentence rewrite has the same referent (The boys / they).