I'd just generate an array containing the numbers 0 .. N, and shuffle it.
use List::Util 'shuffle';
my @random = shuffle 0 .. $N;
The problem with your approach is that not only the chances of misses rises as you have selected more items — it is not even garanteed to ever finish; but also, as you try to take the nearest item to your latest miss, I'm quite sure its randomness will not be that uniform, as items close to clusters of items that have been selected already, have a higher chance of being picked next.
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