Hey there. So a common trick is to use the $/ perl variable. It enables you to read input data record-by-record instead of by line. When set to "" the input will be read as blocks that are separated by empty lines. So, with the input file you provided:

use strict;
use autodie;

open my $FH, '<', 'test.dat';
open my $FH2, '>', 'seq.dat';

for (do {local $/ = ""; <$FH>}) {
    my ($name) = /Name:\s+(.+)/;
    my ($seq) = /Nucleotide\sSequence:\s(.*)/xms;
    $seq =~ s/\n//g;
    print $FH2 "$name\n$seq\n\n";
}
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Produces:

cadherin 4, type 1, R-cadherin (retinal)
atgaccgcgggcgccggcgtgctccttctgctgctctcgctctccggcacagcgagactggagatatcgt
+cacagtggcggctggcctggaccgagagaaagttcagcagtacacagcagcttgcgcatcctgtacctg
+gaggccgggatgtatgacgtccccatcatcgtcacagactctggaaa

tetraspanin 32
atggggccttggagtcgagtcagggttgccaaatgccagatgctggtc

tumor suppressing subtransferable candidate 4
atggctgaggcaggaacaggtgagccgtcccccagcgtggagggcgaa
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