That does not sound good then, the hot shoe does not work like that. The manual flash should fire in sync and I cannot imagine what could keep it from syncing at 1/60 second shutter, or anything you could do to prevent it. We can imagine things if a remote flash, but hot shoe just has the one possibility. You say unaffected, but I assume the flash is actually firing? In a lighted room, it can be hard to tell in the viewfinder, since the mirror raises to close off that view during the shutter.
I'm sorry, I don't know... it does sound like a hardware problem. Some way to try the flash on a different camera, or the camera with a different flash might lead somewhere, at least to confirm the problem.