I think you do have to let the airline know you are not returning on the flight you booked - but can't remember why ...........there was a thread a while back about someone who wanted to fly out to Spain on a package but intended to drive home with friends.
Best to check with the airline/holiday company I'd say.

- maybe they can transfer you to a later flight with another carrier, and/ or extend your hotel stay, for a fee - that could be cheaper than doing it yourself (or at least you would have a price to compare with)??? I take it that the problem is with the return flight not the hotel being full ? Might be worth asking if they can refund any money for the unused flight ?

I'm sure there is someone on here that will come along & help you that is more of an expert than me.