Playing with the input record separator so that you're reading in records where the separator is .slow can simplify things a little. This way you're reading 'records' rather than lines, so you don't have to worry about keeping track of state flags.

use strict;
use warnings;

my $in_file_name = 'sample.txt';

{
    local $/ = '.slow';
    open my $input, '<', $in_file_name or die $!;
    while( my $rec = <$input> ) {
        chomp $rec;
        next unless $rec =~ m/(\d+)\.pace(.+)$/s;
        my( $out_file_name, $capture ) = ( $1, $2 );
        open my $output, '>', $out_file_name or die $!;
        print $output $capture;
        close $output or die $!;
    }
    close $input;
}
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Untested. But I think it gets the idea across.