解:$(1) $设每头牛值$ x $两银子, 每只羊值$ y $两银子
由题意, 得$ \{\begin{array}{l}5x+2y=19\\2x+5y=16\end{array},$解得:$\{\begin{array}{l}x=3\\y=2\end{array} $
∴每头牛值$3$两银子, 每只羊值$2$两银子.
$(2)$设该商人购买$ a $头牛和$ b $只羊.
由题意, 得$ 3a+2b=19 $
∵$a, b $都是正整数
∴$\{\begin{array}{l}a=1\\b=8\end{array}$或$\{\begin{array}{l}a=3\\b=5\end{array} $或$ \{\begin{array}{l}a=5\\b=2\end{array} $
∴该商人共有$3$种购买方法$:①1$头牛和$8$只羊$;②3$头牛和$5$只羊$;③5$头牛和$2$只羊