“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

Solution 1:

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.

Solution 2:

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.