Since the file you are reading seems, judging from the sample, to contain blocks separated by a double newline, you could set $/ (input record separator) to "\n\n" which would give you one <TABLE> .. </TABLE> block as one record, to process that record further as a multi-line string with a m//mg.

But you could also use a flip-flop:

my $table_name;
my $sqldef;
while (<>) {
    if (my $flipflop = /<TABLE name="([^"]+)"/ .. /<\/TABLE/) {
        if ($1) {
            $table_name = $1;
            next;
        }
        elsif ($flipflop =~ /E0/) {
            print "table name: $table_name\n";
            print "sqldef: |$sqldef|\n";
            $table_name = $sqldef = '';
        }
        else {
            $sqldef .= $_;
        }
    }
}
__END__
table name: table1
sqldef: |FieldName1 VARCHAR(20)
FieldName2 INT(20)
FieldName3 BOOL
|
table name: table
sqldef: |FieldName1 VARCHAR(20)
FieldName2 INT(20)
FieldName3 BOOL
|
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See ".." in perlop.