“Will we be able to talk?” I asked, my eyes red and swollen from crying, a balled up tissue squeezed tightly between my sweaty palms

The two clauses with the participles swollen and balled up give cicumstantial details. In the simplest form you could express those circumstances this way: and my eyes were red and swollen from crying and a balled up tissue was squeezed tightly between my sweaty palms.

Generally you can change such long sentences containing circumstantial details into participle constructions. They are shorter and the stylistic effect is more elegant. Of course, this is a literary device of written language.

As Cerberus already mentioned such participle constructions are called absolute constructions. Latin absolutus meant a construction somehow loosened from the main sentence. These constructions can correspond to a main sentence with "and" or a subclause with because, while, when, after etc. Whether the absolute construction indicates a cause or time depends on the content of participle construction and main clause.

In my view these participle constructions correspond to adverbial sentence parts when referring to a main clause. After a noun they refer to relative clauses.

They are called 'supplements'. The first is a verbless clause, cf. "my eyes were red and swollen" ... " (“my eyes” is the subject), and the second is a non-finite clause.

Supplements don't modify anything; instead they have an 'anchor', in this case "I". They are set apart from the surrounding narrative by punctuation, usually commas (as here), and by a slight pause in speech.

Your sentence could be rephrased to the following when you put the dropped being back after each subject of the absolute construction:

Will we be able to talk?” I asked, my eyes being red and swollen from crying, a balled up tissue being squeezed tightly between my sweaty palms.

The full (longer) version before the construction will be

Will we be able to talk?” I asked, and my eyes were red and swollen from crying, and a balled up tissue was squeezed tightly between my sweaty palms.

You could notice that the inflected (past tense of to be) were / was is changed to being to make it a non-finite clause (absolute clause). Then, the being is omitted.

This construction is also known as "participial (participle) construction" and for further information, please visit the linked article which illustrates how it works.

Consider the following sentence:

He walked down the street, his shoelaces untied.

That sentence, from the point of view of meaning, is equivalent to:

He walked down the street, with his shoelaces untied.

although the two sentences differ in that with explicitly links {his shoelaces untied} to {walked down the street} in the main clause. In the first version, that relationship is implicit.

In the first version, the phrase {his shoelaces untied} is an absolute statement, one lacking a tensed verb, and one whose connection to the main clause is implicit, not explicit. The fact expressed in the absolute is understood to be true (or understood to obtain) coeval with the time of the main clause, that is, the time of the tensed verb in the main clause.

After I win the lottery, I will be driving a fancy car through the neighborhood, a happy tune in my head.