Texting is absolutely not part of my unlimited data plan (or anybody else's, I'd be willing to bet).
I've had VZW Unlimited Data for years now. Before I purchased the $5/250 text message plan a few months ago, I was being charged $0.20 per text message (and, $0.25 per message within the last year). I always had a few bucks being tacked onto my bill every month before I bought the $5 plan.
Regarding "free texting": Yes, it is definitely possible. If you register for a Google Voice number, you can send free text messages through GV, and you can also make free (no minutes used) domestic calls through GV. GV becomes a VoIP service that you can make voice calls through the data network. If you register for GV, be prepared to wait a month or so for your "GV Invite". There's quite a lot of demand for GV numbers right now.
That said, making free GV calls/texting also violates VZW's Terms of Use.
FYI,
AzJazz