I do not respectfully understand what some people are doing when grouting. Actually grouting is the easiest part of the tile job and grout only goes in the throats of the tile. If you use that special rubber squeggie to apply the grout and work it in from four sides that should be it. But you do not float the grout on the entire surface.
Then wait 20 minutes and begin the endless task of wiping off the excess grout and haze from the surface of the glazed tile. That is it.
Leaving on grout in clumps and whatever is disasterous and almost impossible to clean up.
Roto-zip, dremel, and hammer and chisel will clean up some, but you take the gamble of damaging the tile with one slip of the tool. Sometimes you have to bit the bullet and remove the work and re do it the right way.
Sorry for your woes, and good luck in the removal, but it may be easier to start over.