Here's my experience texting with AT&T while on the island with an iPhone. If you're not using an iPhone, everything below may not be as relevant.

We could send texts through iMessage while connected to WiFi to other iPhone users. We could not send messages through iMessage to those that weren't also using an iPhone, and iMessage won't work without WiFi or cell data. As in the United States, the messages to non-iPhone users would show up in green and would use the cellular network rather than WiFi (what iMessage uses). Those wouldn't work on airplane mode -- even if connected to WiFi -- because you need the cell network to send a normal text.

So, in my experience, your ability to text without getting charged may depend on the type of phone you have and the type of phone your family has. If everyone agrees to use WhatsApp (or something else), it's one way to get everyone on the same page regardless of the type of phone you're all using as long as you can connect to WiFi.