Betty:
In my own humble opinion, if you were to use a cheap cover, or a tarp on your TM, (or anything else, for that matter), yes, it would hold in moisture and cause damage,,,,but,,,, a "good" cover will breathe. Meaning, it will let the moisture out. The weave is fine enough to hold out drops of water, but open enough to let water vapors out.
That's one reason some covers are so expensve.
Again, not saying you were told wrong, just my opinion, nothing more.
Mack