You can accumulate the current string in a scalar and then print and reset it as soon as you know you'll want to. The code will look a lot like what you have.

#!/usr/bin/perl -w
use strict;

my ($string, $count);

while(<DATA>){
    s/\s//g;
    next if !$count and /^>/;
    if (/^>/ and $count) {
        print $count, ' : ', $string, $/;
        $string = '';
        $count++;
        next;
    }    
    $count || $count++;
    $string .= $_;
}
print $count, ' : ', $string, $/;

__DATA__
> Seq 1 (two lines)
AAAAAAAAAAAAA
CCAAAAAAAAAAA
> Seq 2 (two lines)
AAAAAAAAAAAAA
AAAAAAAAAAAAA
> Seq 3 (one line)
TTTTTTTTTTTTAACTGAAGATTCGC
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I've removed a fencepost error by checking that the > line is not the first.

Having a leading marker instead of a trailing one makes this a little awkward.