Have you searched for a Ferienwohnung (vacation apartment)? You might have some luck there.

Here's a website for Rügen:

http://www.ostsee-reisen.de/ruegen/

There is some availability if you do a search on that site for Rügen. depending on your dates. Let us know if you need help with the German.

Thank you, Woyzeck for the offer of help with German. I use my phrase book when writing and can sort of figure out the answers using a translation website, but it's the telephone where we could really use some help!!

Any suggestions for another resort area that we can try???

If you cannot find anything on Sylt, either look into the neighbouring Danish island of Roem (Roemoe) or, if you indeed want to stay in Germany and consider the Baltic Sea (which is calmer, but has its own charms), take a look at the Fischland/Darss peninsula, just East of Rostock.
It's about an hour closer to Hamburg than Ruegen, although by train and bus it also takes 3.5 hours (still better than wasting half a day on the journey to Ruegen in my opinion).
Ahrenshoop labels itself as artsy and has very pleasant surroundings.
While not an island, the peninsula is quite narrow and you are never far from the water - either the open sea or the "Bodden", the enclosed backwaters on the landside of the peninsula.
There are other resort towns at the Baltic Sea much closer to Hamburg. The Western ones are fairly developed and tend to be crowded, which is not to say that they do not have their charms. Timmendorfer Strand at 45 min. distance from Hamburg is probably the favourite destination for younger people and is slightly more up-market than neighbouring destinations.

I would look into Timmendorfer Strand et al. mainly for daytrips, by the way. At such close distance it may not necessarily be worth spending top-of-the range EUR on accomodation if you can reach your destination in about 1 hour.
Fischland/Darss should be about a 3-day trip, though.
Again try
http://www.hrs.de
(switch to English in the upper right hand)
as it lets you search the surroundings, too.

hsv: Thank you once again. You have given us some great ideas for day trips.

We were able to book at the Lindner Hotel Windrose on Sylt. It is more money than I planned on and I am not sure about the location. Also, it looks like a huge corporate enterprise and I usually prefer independent hotels. However, if I can get my husband to agree, we might take it if its location is good. So, hsv and Fodorites, can you tell me if this hotel is worth the Euros and if its location is good? They are quoting us 564.30 Euro for 3 nights including breakfast. We will not have a car. Thanks again.

I don't know the hotel.
Lindner is a very mid-market chain in Germany. I stayed at one of their hotels once. It was clean and very ordinary. It wasn't on Sylt, though. If the one on Sylt is anything like my experience it is alright, but doesn't necessarily capture the spirit of the island.
Wenningstedt is pretty family oriented and very mid-market itself. It's not far from Westerland and the station and the red cliff close to Kampen in the other direction.
I guess overall it's alright - but is the hotel worth the money? I seriously cannot tell you without knowing the place.
I hope others can help out.

Thank you, hsv.

What about Hiddensee Island? We can stay at a small hotel in Vitte for 95 Euro. Would that be a better choice than the Lindner? Thanks.

Sylt is about 3 hours by train from Hamburg.
Hiddensee is about 6.5 hours by train from Hamburg. To me that is a lot of travel time that is pretty much wasted. Back and forth it accounts for a whole day. And actually with 6.5 hours on a train there isn't that much of a day left after your arrival (although the long summer days make it more bearable).
Hiddensee is too far for me - I cannot advise.
If I were you I'd go for Sylt at the North Sea or on the Baltic Coast the Fischland/Darss peninsula. Saves a lot of travel time and both are nice enough.

Thank you, hsv. I agree with you, Hiddensee is probably too far. I'll get my husband's take on it.

And Hiddensee is tiny. Sleepy. Indeed too far. Wouldn't be a good choice.

I had hoped you would be able to add some colour on Hiddensee, Ingo.

"Hoch stand der Sanddorn am Strand von Hiddensee..."

Thank you, hsv, Ingo and Woyzeck. WE have decided not to take the room in Hiddensee. My home exchange concurred that it would be too isolated. She also, gave me some telephone numbers and websites that I will try tomorrow. The search continues. . . I am optimistic.