These songs were not sung to the tune of te'amim (trope is Yiddish, so let's bring it all back to Hebrew so that all Jews can feel included :D, or use the English word of Cantillation)
While there are varying beliefs on how/when the Biblical books were written, i think most people would agree that the Prophetic books were not written before the actual events. So in other words, when Devorah sang her song, it was written down AFTER it was already a song. Therefore it would be a stretch to have the song's development be limited to te'amim.
The same is with the song of the sea by Miryam. Even the strictest believers to the Rambam's 13 principles of faith would say that Moshe received the Torah at Sinai. Which means that since Miryam sung her song at the sea before they reached Mount Sinai, her song also existed as a musical piece before Moshe wrote down the Torah, and therefore it would be unlikely that it was sung according to the te'amim.
In the case of Dawid's psalms, they were often designed to be played with musical instruments, and to be sung by groups of people. Since te'amim are not played with musical instruments, and are not done with groups of people, it makes the most sense that the psalms were not sung according to the te'amim.
Also, since the majority of the readers here are of Ashkenazi origin, a few notes will have to be made.
The Ashkenazim are unique in having a very long tradition of having all of their te'amim be musical. The rest of the Jewish world did not have this development until very recently. For the rest of the Jewish world, most of the te'amim symbols have no musical value, they only serve as grammatical indicators for readers. The Yemenites to this day have only 8 musical motifs for their entire cantillation system. The Karaites also don't have a musical tradition for all the te'amim til this very day.
In the 14th century it was noted by Simon Duran in his book Magen Avot: "For the teren chutrin the same motive is used for the tevir, except that it is considerably lengethened, but we know no musical difference between the two...we [also] know no difference in the melody of the revia, qadma, azla, gershayim/shnei gereshin, and zaqeif gadol."
Even into the 1500s it was noted by the scholar Elias Levitas in his book Tuv Te'am that only the Ashkenazim had musical motifs for all the te'amim.
Image from Idehlson's Thesaurus of Oriental Hebrew Melodies
Since it's most likely that most of the te'amim were not originally musical, nor were ever intended to be musical, it would stand to reason that ancient Biblical songs would not be based on a mostly non musical system.