What if you try this?

my $SID=shift;
my $output=`. /my_location/switchdb.ksh $SID 2>/dev/null 1>/dev/null &
+& echo \$ORACLE_HOME`;
chomp($output);
if($output =~ m{/}){
  $ENV{ORACLE_HOME} = $output;
}
print "DEFINED ORACLE_HOME as (" . $ENV{ORACLE_HOME} . ")\n";
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Note that I do not execute, I source the script. Then request the variable with and (&&)

in a similar fashion, if you need more environment variables, dump them with env:

my $output=`. /my_location/switchdb.ksh $SID 2>/dev/null 1>/dev/null &
+& env > /tmp/$mytempfile`;
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then read the file back and split(/=/,$line,2) to get the key and the value, and put them in %ENV