Most ashkenazim treated Shabbos the same as in Chu”l, not eating chametz or kitniyot. Some ashkenazim ate kitniyot. After all, it was not Pesach anymore, and kitniyot is not chametz and is not sold to the goy.
Soft matzah is really irrelevant to this discussion. It is matzah like any other matzah except that it is thicker and soft. It can be eaten at the seder and through Pesach. Many ashkenazim have an emotional aversion to it because it looks too much like an Iraqi pita or laffa, but that’s purely psychological. After Pesach, this psychological barrier falls and they eat it.
To each his own.