If my son doesn't get married, the Sato family name will be cut off.
If my son stays single, our family line will not continue.
この場合、次のように言えますよ。
ーIf my son doesn't get married, the Sato family name will be cut off.
「息子が結婚しないと、佐藤家は途絶えてしまう」
the XX family name will be cut off で「XX家は途絶えるだろう」
ーIf my son stays single, our family line will not continue.
「息子が独身のままだと、家系が途絶える」
our family line will not continue で「家系が途絶える」
ご参考まで!