A general FASTA-processing loop looks like this:

# Always read up to the next ">" at the beginning of a line:
$/ = "\n>";
getc(INPUT);                    # skip the first ">"
while (<INPUT>) {
    # remove "\n>"
    chomp;
    # first line is the ID, the rest is the sequence
    my ($id, $seq) = /^([^\n]*)\n(.*)/s;
    # remove whitespace in the sequence
    $seq =~ s/\s+//g;
    # do something with $id and $seq
    ...
    # output the result in 72-column format
    print OUTPUT ">$id\n";
    print OUTPUT "$1\n" while $seq =~ /(.{1,72})/g;
}
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You'll want to read and write the data as you go instead of storing it in an array, and check for all the primers in one pass.