it seems that the origins of 'schtuk' are lost in time, but it is a slang expression meaning the result of a failing or error - so a different word that 'stuck'.
Schtum has its roots in the German language, but the pronunciation seems to have evolved in English.
In common with most of my generation, I learned the word from Monty Python, which was a veritable oasis if culture - palindrome anyone?
The word is not 'stum' , if you mean 'not speaking' , 'mute'. It's 'shtumm' , sometimes written 'shtoom', a Yiddish word (from German 'stumm').
"I'm in shtuck" is also correct. 'Shtuck' or 'shtook' is a word meaning 'trouble' . The speaker is saying that he or she is in trouble. This may, perhaps, be from some local form of Yiddish or it may be imitating Yiddish , but the Oxford English Dictionary says it's origin is unkown.
In general, German words starting with "st" or "sp" are pronounced "sht" and "shp". So the magazine "Der Spiegel" (The Mirror), is pronounced Der Shpeegel and the word which means "you can say that again, "stimmt" is pronounced shtimt.