2 things:

1. Yes, but what you don't know is whether the same token has been given to someone else.

2. But if you (or your wallet) provides a public key, then the ecash can be locked to the public key and then you can be sure that you have cash with your name on it™ and can validate it offline. (I've never done this, not sure if it's a released feature yet but it is definitely possible)

With 1. You need to claim your ecash immediately for sats or for a fresh ecash to prevent double spending.

With 2. The sender either needs to prepare the ecash with your name before they send it to you while they have internet, or they need to have internet when they send it to you. You as the receiver don't need internet to prevent double spend in this scenario.