One sudo for multiple terminals

The -n option is available in newer versions of sudo, but as you stated that's not an option. There's no real way to do what you're looking for short of just trying sudo and seeing if it comes back with a prompt for a password. If your concern is you want a visual indication, why not start do sudo /bin/bash to start a root bash session? Note that this is insecure, but it's also somewhat insecure if someone realizes your prompt changes on sudo.

Try adding this to your sudo options:

Defaults timestamp_timeout=0, tty_tickets

tty_tickets option (on by default) will make sudo ask password if it was not asked previously in that particular tty (including terminal emulators ptys), and timestamp_timeout=0 option will make it not ask it again in the whole session.

So, when you want to do some administrative operations, you can open terminal, sudo something, close it, and you will be safe again.