***** NOT A TEACHER *****
(1) According to Professor George O. Curme in his great
two-volume A Grammar of the English Language, your
sentence is an example of ellipsis.
(a) He gives these examples:
(i) She does nothing but [that she does] laugh.
(ii) I'll do anything to show my gratitude but (or except)
[that I do] marry your daughter.
(2) So I guess the full sentence of your first elliptical
sentence is something like:
I could do nothing but that I could wait.
(3) Regarding your second example, I have less confidence
in guessing. Maybe (a big "maybe"), it is SOMETHING like:
There was no other choice but that I did tighten my belt.