A paean is one of the three main genres of religious songs (the others are the hymn and the dithyramb). We do have the texts of several paeans, but they were written for religious festivals, rather than specifically for battle situations. As well as those two settings, they could also be sung at a dinner or symposium, or in a wedding procession. They were supposed to have a calming effect, associated with equanimity and serenity. (There's one story of Pythagoras using paeans for a calming effect on his cult members or to heal diseases, but that's only attested very late, and is based on a wordplay between paian and an early god of healing, Paiēwōn.)

They varied a lot in scope. They could range from a simple repetition of the phrase iē Paian, iē Paian to something like Pindar's paeans (early-mid-5th century BCE), which are long, involved compositions, of a similar scope and attention to detail as a Beethoven sonata movement.

What did they sound like? Well, they could be sung either solo or in a unison chorus; the ones you've been reading about were of course choral. Accompaniment would vary depending on circumstances. We actually have substantial chunks of the scores of two paeans composed for performance at Delphi in 127 BCE: here's a transcription of one of them, by Athenaeus, in modern notation. Be warned that the pitches do not correspond at all exactly to the temperament system used in modern western music; however, enough is retrievable to see that this paean used a pentatonic scale.

Another warning: extant scores of ancient Greek music are very basic and not exactly representative of actual performance styles. They don't give any indications of things like rhythmic and melodic ornamentation, and those are important things: in modern music, they're the difference between a Grade 3 fingering exercise and Frank Sinatra. So we can't be at all confident about how an actual performance would sound.

Also note that if the sources you've been reading relate to events prior to 400 BCE or so, musical styles prior to that date were very different and we know even less about them than we do about the kind of music composed by Athenaeus.