Just a few things from skimming over your code:

Use Placeholders!.

Instead of

my $sql = "INSERT INTO ERRORS (TSTAMP,CMND_ID,ALERT_ID,SEND_ID,DESTIN
+ATION,MESSAGE,STATE,SVR_HOST) VALUES (\'$tstamp\',\'$cmnd_id\',\'$ale
+rt_id\',\'$send
+_id\',\'$destination\',\'$message\',\'$state\',\'$svr_host\')\;\n";
                my $statement = $db_handle->prepare($sql)
                    or die "Couldn't prepare query '$sql': $DBI::errst
+r\n";
                $statement->execute()
                    or die "Couldn't execute query '$sql': $DBI::errst
+r\n";
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write

my $sql = qq{"INSERT INTO ERRORS (TSTAMP,CMND_ID,ALERT_ID,SEND_ID,DEST
+INATION,MESSAGE,STATE,SVR_HOST) VALUES (?, ?, ?, ?, ?, ?, ?, ?, ?);
my $statement = $dbh->prepare($sql);
$statement->execute($tstamp, $cmnd_id, ...);
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Also make sure to pass the {RaiseError =>1} argument to DBI->connect, then you don't need the custom error handlers.

Even more important: you can prepare the statements once outside the loop, and then call execute them inside the loops as many times as you need them. That way you'll gain much performance (if you have many iterations).

Instead of

if($line =~ /^SendCleared~(.*)~(.*)~(.*)~(.*)~(.*)~(.*)/){
                my $tstamp = $1;
                my $cmnd_id = $2;
                my $alert_id = $3;
                my $send_id = $4;
                my $state = $5;
                my $svr_host = $6;
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you can first check if the line starts with SendCleared and then use

my ($tstamp, $cmnd_id, ...) = split m/~/, $line;
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There are actually even better ways to handle the dispatch with a hash: you can store these sql queries in a has with keys SendOffDuty, AlertCmd and so on, and then do something like that:

# assume %q contains the quries
my ($query, @args) = split $line, '~';
$q{$query}->execute(@args); # assuming the log files are well formed
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