I think they're both technically correct but it depends on where you live (especially in the US), I'm from the northeast and we rarely say "what have you got to do right now", we'd say the 2nd sentence you suggest. I think in the southern us they use "have you got to do" more often. To me "you got" sounds strange even though it's not technically wrong.
I've had the same issue with the female, as well as the male, voice running words and sounds together. SO...First I listen to "her" and type my response, but before I send it to "her", I listen to the slower speed and I adjust my answer accordingly. Sometimes I forget that final step, and I curse like a sailor!