Your transfer switch is designed to prevent conflicting issues between the two sources of AC power. It is designed to only allow power from one source or the other, and won't let them both connect at the same time. It probably also has a delay built in so if an air conditioner is on, it won't restart the a/c until pressures even out and cause stalling if the compressor.
GFI circuits are notoriously sensitive, which is a good thing to prevent a shock hazard. They trip if any imbalance between the hot and neutral wires is detected. Does the GFI trip if you plug in to shore power and don't have generator running? If so, you'll need to check that your shore cord plug and outlet are properly wired. Then turn off every circuit breaker in the trailer and plug it in to shore power. One by one turn on circuit breakers and see if the issue is in one of your circuits.