Your code will only find the 2nd string if it is on the same line as the 1st string: foo 002389983 blah 29994339499

Stop looking through the file as soon as the 1st string is found. If the 1st string is found, start looking through the file from the beginning again. Stop looking through the file as soon as the 2nd string is found. Untested:

my $found_first  = 0;
my $found_second = 0;
while (<LOG>) {
    if (/002389983/) {
        print "Looking for: 002389983\n Found it in file: $_\n";
        $found_first = 1;
        last;
    }
}
if ($found_first) {
    seek LOG, 0, 0;
    while (<LOG>) {
        if (/002389983/) {
            print "Looking for: 29994339499\n Found it in file: $_\n";
            $found_second = 1;
            last;
        }
    }
}
unless ($found_second) {
    print "Could not find the second string: 29994339499\n";
}
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