Translation:I am looking for water. It is necessary for my dog.
I'd say that there's nothing negative about "szukać" (although some people use a mnemonic technique and consider that if you're looking for something, it means that right now you don't have it and that's negative). The verb "szukać" is just one of those exceptions that take a direct object in Genitive instead of Accusative, you need to remember those.
"mojego psa" is in Genitive because of the word "dla".