さもないと has a very restricted and loaded usage, and not technically a conditional. It's only used in situations such as "Otherwise / On failure, [here come some bad consequences]".
しないと and でないと are respectively...:
- しないと =
する ("do") + negative +
と ("if/once...") "if (something) doesn't do"
- でないと =
だ／である (copula) + negative +
と ("if/once...") "if (something) is not"
But as you said, some phrases starting from a copula are allowed to be used in the beginning of a sentence as conjunctions, as if the entire previous context connects to them: だが, だけど, だから, でも etc.
As such, でないと (and its colloquial form じゃないと) can stand at the beginning of a sentence to mean "if it is not the case" or "otherwise", but しないと just means what it would.