Don't worry, your tent is waterproof.
Your tent is waterproof, but the moisture on the inside of the tent in the morning is condensation. Everyone who sleeps exudes moisture during the night, which hits the wall of your tent and condenses.
This happens faster when it is cold outside.
If all the windows and doors are closed, the humidity of the air inside the tent increases.
At night, and especially at low temperatures, moisture from that air condenses on the coldest surfaces. Most of the moisture settles on the tent fabric. If there is a lot of it, it even drips down and wets your bedding. It is the same as what happens to the mirror in the bathroom that fogs up when you shower or take a bath. It is colder than the air in the bathroom. And therefore the moisture from the air condenses on the cold surface of the mirror.
