Hello,
Without regrouting it will be difficult for new grout to bond copletely. If you want good long term results copletely remove old grout first.(Dremel makes a fairly handy attachment for their rotary tool,
You may want to consider calling local stone restoration company. They have extensive experience and the cost is modest. Your project will be finished in timely manner and
you don,t have to spend the time down on hands and knees.