One DOES have to have a good ear and an excellent set of earphones to hear "Shoot me", but it IS there. John understates the "me". I'm sure the isolated vocal track would be the ultimate proof. I can't say I heard it in the years immediately following its release. I heard "shoo" or maybe "shoop", but that was on vinyl. If you have it on CD and can play it on a standard CD player (not on iPod thru earbuds). fiddle with the stereo separation and treble/bass. You'll hear it.
Yes, John did seem to have a premonition of his death by murder, according to several biographies. This may have turned into a preoccupation. The three Beatle/Lennon solo tracks that stand out are: "Come Together"; "Happiness Is A Warm Gun"; and "Whatever Gets You Through the Night" ("Don't need a gun to blow your mind").
But - and this is the most amazing part - if you go through Lennon's entire work, there are dozens of examples of references to words such as, "die", "dead", "death", "suicide", "kill", "gun", and so forth. John did have a morbid sense of humor, no doubt. I'm no psychologist, but I do believe lyricists have a way of revealing themselves, and John had more opportunities than most.
John also had (though not as much as George) associations with "spiritualists" with strong Buddhist/Hindu philosophies. Many of them might be labeled "mystics". Some of these mystics believe that a soul enters a new life already subconsciously knowing its ultimate fate. I'm not suggesting you believe as they do, but could John have come to know his own fate as a result of these associations? John has been quoted as saying that some of his intuition about his future had been confirmed through meditation, though he didn't mention any death premonition.
And if a "psychic" did predict his death, would Lennon have believed it or dismissed it? Read Larry Kane's book about the 64 US tour to find out the reaction to Jeane Dixon's prediction that the band would die in a plane crash. John took it seriously. George went so far as to call Jeane Dixon for confirmation. Later, John would say he was obsessed with Buddy Holly's death at the time.
And so, Beatle People, I offer all the above FYI. Does it prove anything? Nope. But is IS interesting.