Our new bathroom fan is not clearing the room the ceiling is wet and the paint peeling the fan is model #FV-LP003

I use to be a chimney sweep. One of my jobs was to clean vents for dryers and bath vents. What could be your problem is that you have a bird nest in the vent system. It sounds to me that you have a vent on the roof for the fan. A bird could have damaged the cap to gain entrance and built a nest. With the cap gone, rain can come in. With the nest, it will not vent out the room. You could call your local chimney sweep company and they will clean it for you. Or if you are capable, Lowe's or Home Depot sell dryer vent cleaning rods. They are sturdy but flexible rods with a 4" brush on the end. Two other handy tools will be a high power shop vac to **** out anything from inside the vent, and a high power blower that has vacuum capability to use out side the house. Cafe full though, if it pulls out a nest, it will shoot it out with great force.