When your fan runs, it pulls a suction (vacuum) on one side of the fan (the return air side). This suction can hold up water from draining. When the fan stops, the vacuum goes away and all the water that's been held up drains out. With low humidity, you may not get enough water to see this effect, but with high humidity you would. If this is happening, there's a problem with your drains.
(The other problem with this is that, since no water ever drains out while the fan is running, you aren't really removing any moisture from the air in the TT. We had this issue in a building I once worked in. Couldn't figure out why the AC wasn't removing moisture from the building. Turned out the "drains" were just a hole on the suction side of the fan. The water couldn't drain out due to the suction. Fix was to put a 6" long "J" shaped drain attached to the hole. The 6" of standing water overcame the fan suction and the water could drain out.)
It's certainly possible that when it rains, you have a higher relative humidity, and thus more moisture for the AC to remove from the TT air, and hence more water being held up by the vacuum I described above. But I don't see hard rain necessarily being worse than a light rain in this case (it would be the actual relative humidity, which isn't necessarily directly proportional to the rate of rain fall. Therefore, I'm still not convinced that you don't have an external leak on your AC unit above or at the roof. In fact, you seem to have improved the situation with your bolt tightening, which certainly makes the roof seal suspect.
1988 Coleman Sequoia - popup (1987-2009) - outlasted 3 Dodge Grand Caravans!
2012 Roo19 - hybid (2012-2015)
2016 Mini Lite 2503S - tt (2015 - ???)
2011 Traverse LT, 3.6L, FWD
2009 Silverado 1500 Ext Cab, 5.3L, 4x4, 3.73
2016 Silverado 2500HD Dbl Cab, 6.0L 4x4, 4.10