We usually don't pronounce "wadashi", we pronounce "watashi" basically. And we also say "hajimemashite", not "hajimemashide". I think you heard TOHOKU dialect. TOHOKU is north part of Japan such as AOMORI, IWATE, and AKITA. They sometimes pronounce "wadashi" instead of "watashi".

You might have problems with this because Cantonese (which I'll assume is your native language) does not have the /d/ sound. In Cantonese, D is a non-aspirated T. In Japanese it's a voiced T. http://vocaroo.com/i/s1MiNMJ0I7sU - Voiced, Japanese D http://vocaroo.com/i/s1OxhPSGDgOE - Unvoiced, Cantonese D, Japanese T In Japanese there is no differentiation between an aspirated and non-aspirated T. So D and T in Cantonese are both T in Japanese