There are quite a number of songs that legimately belong to the Revolutionary War period and earlier, some very beautiful.
An example is "Over the Hills and Far Away."
Look at the list on Contemplator- not all are verified, but there are many good choices. Popular songs in American History

If you like 'Siul A Ruin' in its many versions and wish to presume its existence in the 18th century, go ahead, but there is no evidence of its lyrics at that time, especially not in America. The tune, perhaps, but not the lyrics.