Possibly connected or just similar, drache means dragon in nearby German.
Walloon: pour with rain, from the verb dracher. https://translate.google.co.uk/translate?hl=en&sl=fr&u=http://chanae.walon.org/lh/wa/dic/index.php%3Fquery%3Ddrache&prev=search
Old English: Refuse https://books.google.co.uk/books?id=HQhaYTFaAz0C&pg=PA27&lpg=PA27&dq=drache+walloon&source
so clearly each country has an alternative unrelated source neither meaning the same as your words.
Here is an analysis of the Walloon version: http://phlegmish-walloony.blogspot.co.uk/2014/09/belgian-weather-in-nutshell-la-drache.html
which also extends the word into Flemish as den drache.
Dreich is old German/English or Brittonic drycin (via a common root) so has clearly changed meaning since then, and is also Irish Gaelic. https://en.wiktionary.org/wiki/dreich#Etymology
So we know dreich is via Germany and England, while drache, while sharing the meaning, although the dragon now comes close to the water with ducks, as the next similar root is drake. Drache in fact is also Germanic with many variations
Drake again, along with Drage, and the French Drache, Dracq, is most probably from a root drac, drag, trag, found in many Old Germ, names...
So dragon/drake appears to be the initial meaning, mutating into drake (wet), and spreading north across NW Europe. It is hundreds of pages long so requires some future searching.
Ultimately it would be highly unlikely for them not to have a shared root, but tracing it is not quite conclusive yet.