I prefer to record the positions of the single quotes then use substr to escape them, working backwards from the end of the string. I tend to avoid double backslash confusion by using the \x5c ASCII literal, although in this simple case q{\\} would be fine.

use 5.026;
use warnings;

my $string = q{This sting it's bothering to escape ' single characters
+};

my @posns;
push @posns, pos $string
   while $string =~ m{(?=')}g;

substr $string, $_, 0, qq{\x5c}
   for reverse @posns;
say $string;
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I hope this is helpful.