Downgrading to 1.3.16 (184.108.40.206169) seems to have fixed the hanging issue, but has introduced an issue where files deleted on one machine are not being removed from another. GDrive reflects the deletion, as do other apps that sync with GDrive, so it's isolated to InSync.
My biggest concern at this point is exposure to data loss because of unstable InSync coding. Please let me know if there's an even more stable option, even if some features are lost in the downgrade.
EDIT: I should note that the status indicators (errors, progress, green checks, etc.) are all ok on the machine having the issue.