I think it's just in North Wales (where I'm from) but yes, although it's more complicated than that. See the following link... http://www.bbc.co.uk/wales/learnwelsh/pdf/welshgrammar_ff_replies.pdf
Sometimes it's short for ydw, sometimes for rydw, so you're best bet is just stick with dw!
I'm getting the translation wrong but I don't see the correct sentence written down anywhere. Am I missing something?
The literal translation here is:-
Ydw = 'Yes/Yes, I do'
Dw i = 'I am'
Eisiau = 'to want/wanting'
Cath = 'Cat/A cat'
So this gives:- 'Yes, I am wanting a cat' which is contracted to 'Yes, I want a cat'