Pretty much the entire state of Schleswig-Holstein is worth a closer look.
This site is an excerpt from a book on birdwatching hotspots in Northern Germany (the site is in German, but it has maps and lists individual species):
http://birdinggermany.de/nordsee.htm
All of the highlighted areas on the map are good places to see shorebirds.
Closer to Lübeck around the Baltic coast, there are a few places, most notably the area around Hohwacht (particularly the
Sehlendorfer Binnensee reserve immediately to the east of the town), but also the reserve at
Schmoel:
http://birdinggermany.de/schmoel.htm
The
Wallnau reserve on Fehmarn is about 1 hr from Lübeck, should be very good for shorebirds though. Also requires a car.
Next to Lübeck, there is a place that has a feral population of Rhea (i think around the
Wakenitzniederung).
Then there is the lakelands around Plön (between Lübeck and Kiel), which are mainly good for seeing White-tailed Eagle, Common Crane, and various ducks and grebes (and songbirds). For instance, the
Pohnsdorfer Stauung near Preetz has a population of Greylag and Canada Geese as well as Lapwing and Little Ringed Plover, and a variety of ducks. It's also a great place to see Thrush Nightingale and River Warbler in spring, and WTEs throughout the year. There may also be random waders there on migration.
There's also the
Geltinger Birk near Flensburg, but it's practically in Denmark (2 hrs ride from Lübeck) and is not really suited for public transport.
If you're not afraid of long rides, you can also check out places in Mecklenburg-Vorpommern, e.g. the
Vorpommersche Boddenlandschaft National Park or even
Hiddensee. Going to those places might mean you have to stay overnight somewhere, though.
Regarding public transport, theoretically all places I've mentioned are accessible in some way, but some are only really feasible if you take a bicycle with you on the train. Generally speaking, I'd recommend using a car if possible, especially for the places around the North Sea (e.g.
Katinger Watt), or Schmoel or Pohnsdorf.
edit:
Regarding Heligoland, maybe someone who has been on the island during summer can elaborate on what one could reasonably expect there in August. I've only been there once, in spring. Basically, there are two main birding seasons on Heligoland - spring (from mid-to-late March till June), when the seabirds are breeding, and autumn (especially September/October) for the autumn migration, which is the best time to find vagrants from places as far away as Asia or North America.