I thought I was doing a fairly good job of that, but somehow I've been falling down anyway...

[Which he's doing now, looking more at the ground while he walks than up at Vash. He doesn't want to think he's so clumsy that he practically needs an escort to walk anywhere, but that's about the only explanation for this right now, right?]

Perhaps the snow from yesterday froze over, so the snow for today is sitting on top of ice? That would explain the slipperiness.