Instead of "Where am I going to put my belongings?" is it possible to say "Where my going to put my belongings?"
My is not an abbreviation for "am I".
In Old English "My" and "Mine" were the same word, (min) But by 1200 a reduced form "mi" later "my" was used when the next word started with a consonant. And by 1500 "my" was used for all words. It is completely unrelated to "am I", which in Old English would have been "eom ic"
In some dialects, "am I" can be slurred and might sound like my.
Wher'm I gonna put my belongings.
This is typical of casual speech in some dialects, but would not normally be a style that learners should try to copy.