I got a new Apple Watch, should I unpair my existing one before trying to pair the new one?

If you want to keep using the old watch (as well as the new one), you don't necessarily have to unpair the old one. But if you only plan on wearing the new one you might as well unpair the old one. You can follow the steps in the 'Before you sell, give away, or trade in your Apple Watch' section of:

What to do before you sell, give away, or trade in your Apple Watch, or buy one from someone else - Apple Support

and then you can sell or give away the old one. You can have multiple watches paired to a single iPhone, but not the other way around. See step 4 in:

Set up your Apple Watch - Apple Support