Are absolute phrases adverbials?

Can we say all absolute phrases function as "adverbials" modifying the subject+verb of the sentence that they are attached to?

For example:

Her determination stronger than ever, Nexisa resolved not to give up until she had achieved her dreams.

Does the absolute phrase in italic describe the subject as performing the action?

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).

Instead of the "absolute phrase" you can choose a subordinate clause with an appropriate conjunction, eg "As her determination was stronger than ever, Nexissa ...".

I would say that the entire main clause—"Nexisa resolved not to give up until she had achieved her dream"—is being described or modified by "Her determination stronger than ever." So it's explaining to you about how, in what condition or circumstances, or in what manner she comes to that conclusion. It's almost like an adverbial phrase, but it's not called adverbial phrase because it starts with "Her determination" as the subject of that absolute phrase. Also, an adverbial phrase modifies only the adverb or verb, not the entire clause.

I hope that makes sense.