Is my hydrangea dying?

I received one of those pretty hydrangea plants with the foil around the base as a gift, and it had a very beautiful show, and then the flowers died (as flowers tend to do) which didn’t really bother me. But then the leaves started to fall off, which I became a little concerned about. To give a full picture, I I rent, and I have to keep it indoors because I don’t have any “outdoors”. I keep it right next to a window, and most of the leaves that fell off were on the side away from the window. I have noticed a fine mesh that looks like spiderwebs on it, but it only occurs around the stubs where the flowers were, and its not really in a web that looks useful for catching bugs at all, it looks more like tiny tiny string wrapped around the dead stems of the flowers. I read up on the hydrangea society website about them, and they said not to overwater, so i’ve been watering it a little less, and it seems better than before, but I’m really weirded out by this cobweb thing. Does anyone have a hydrangea? Has anyone seen this before? What can I do about it? Should I do anything about it?