I checked whether the OPs program uses any arguments - he didn't, so exec $0 should be fine. As for context, most context, like environment variables, cwd, and even open file descriptors will be preserved. State information of the process itself of course doesn't, but I already mentioned that. Nor does the OPs program use state information (although the module it uses builds up a state, which causes the memory leak - the state isn't used; the fact the exec loses this state is exactly the reason why exec helps here).

As for preventing duplicates to be run, something like:

use Fcntl ':flock';
open my $me, $0 or die;
flock $me, LOCK_EX|LOCK_NB or exit;
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near the beginning of your program usually does the trick.