The spots do look like a fungal issue to me. How often do you water? If the stem is going black very early, it could be bulb rot as well.
Individual flowers will thin and fall apart after 5 to 8 days, as they generally only last about a week. If they are only lasting a couple of days, though, this would also suggest a problem.
Generally Liliums aren't happy if you transplant them whilst in leaf (with certain exceptions), so if you dug them up to check the bulb and roots this may have damaged the root system and the stem will start to die off as a result. This won't kill the plant, but it will set it back a bit whilst it recovers.