You can do this without any external commands. Define two macros
containing = $(foreach v,$2,$(if $(findstring $1,$v),$v))
not-containing = $(foreach v,$2,$(if $(findstring $1,$v),,$v))
LIST := a_old_tt x_old_da a_new_da q_ty_we
LIST_OLD := $(call containing,old,$(LIST))
LIST_NOT_OLD := $(call not-containing,old,$(LIST))