Взяв би я бандуру
It shares a melody with "Hymn of the Fallen Legionaries" - Simion Lefter. This one is difficult to confidently conclude which melody came first, but evidence is pointing to Ukrainian origin.
The earliest example of the modern lyrics I can find is from 1927, but there are records of similar songs going back into the mid 1800s, and the modern melody works perfectly with those older lyrics. The earliest recording I can find, from 1935, uses the modern melody. The modern music was already published in a spivanyk in Kyiv in 1931 using identical lyrics to those in an УСС spivanyk from 1921.
While I don't have hard confirmation, it is unlikely that the lyrics suddenly switched to a Romanian melody for everyone across Ukraine for multiple very different sets of lyrics, while simultaneously losing some original melody which I cannot find any record of (although it wouldn't be unprecedented, e.g. Гоп стоп канада). I believe this melody was borrowed by Lefter.