The following will work even if you give it only a single line of input. The "flag" you mention is more properly referred to as a state variable.

#!/usr/bin/perl

use strict;
use warnings;

my $retain = "";

while (<DATA>)
{
    chomp;

    if (m{^(<p>)(.*)(</p>)$})
    {
        my ($begin, $middle, $end) = ($1, $2, $3);
        # This line is inside a <p>...</p> match.
        # Handle differently if it's the first line.

        if (! $retain)
        {
            # Found first <p> in block, replace with <con>.
            # Print beginning and middle, but retain end, in case
            # this one line is also the end of the block.
            print "<con>"."$middle";
            $retain = $end;
        }
        else
        {
            print "$retain\n";          # Finish off previous line
            print "$begin"."$middle";   # Start new line
            $retain = $end;             # retain ending
        }
    }
    else
    {
        # This line is not a <p>...</p> match,
        # so flush preserved output, if any,
        # and copy this line to the output.
        if ($retain)
        {
            print "</con>\n";      # Finish off previous line
            $retain = "";
        }
        print "$_\n";
    }
}

if ($retain)
{
    print "</con>\n";      # Finish off previous line
}

#<p>This is para2</p>
#<p>This is para3</p>

__DATA__
<root>
<p>This is para1</p>
</root>
[download]

Input cases and appropriate output:

Case 1 Input: Single line

<p>This is para1</p>
[download]

Case 1 Output:

<con>This is para1</con>
[download]

Case 2 Input: Multiple lines in block

<p>This is para1</p>
<p>This is para2</p>
<p>This is para3</p>
[download]

Case 2 Output:

<con>This is para1</p>
<p>This is para2</p>
<p>This is para3</con>
[download]

Case 3 Input: non match lines present

<root>
<p>This is para1</p>
<p>This is para2</p>
</root>
[download]

Case 3 Output:

<root>
<con>This is para1</p>
<p>This is para2</con>
</root>
[download]

Case 4 Input: Multiple separate blocks (I assume you want separate blocks treated separately.)

<root>
<p>This is para1</p>
<p>This is para2</p>
</root>
<stem>
<p>This is para3</p>
</stem>
[download]

Case 4 Output:

<root>
<con>This is para1</p>
<p>This is para2</con>
</root>
<stem>
<con>This is para3</con>
</stem>
[download]