Nah, it’s OK. I’m honestly not sure if I learnt the contraction from Tobira. Let me go look again…
EDIT: OK, I’m back. I’ve found it, but it’s not explicitly taught. Flip to the next page and look at the table under くだけた話し方. You see 分かんない? What do you think that means? (I’ll let you look that up while I type some explanations/corrections related to your post.)
OK, I’m fairly sure that, perhaps like you, I learnt しなきゃ as a contraction of しなければ thanks to Tobira. I can’t find the page in Tobira that specifically mentions it though, so I’m just going to explain from my own knowledge. (You’ll probably find the relevant section later, if you haven’t already.)
Technically speaking, しなければ just means ‘if not do’. Nothing more. It’s only when しなきゃ (the contracted form) is at the end of an informal sentence that you infer the full phrase, しなければ ならない・いけない, which is literally, in this context, ‘if [I] don’t do [it], it won’t become/go’ = ‘if [I] don’t do [it], it’s unbecoming/it won’t fly’.
Given what I just said, and also given what Tobira shows about 分かんない (which is just an even more informal version of 分からない) in that table, なんない=成んない＝成らない＝ならない. See the connection?
Bonus about the difference between ならない and いけない in this structure: not sure how strict the rule is, but generally, ならない is more for external, social/moral obligations. You know, like how we always say something is ‘unbecoming of [a type of person/organisation etc]’? It’s about what’s expected of them, right? Whereas いけない is for an obligation/necessity felt by the individual on a personal level, maybe because he or she has standards that require action, or because the action is necessary for obtaining something he or she wants.