I was told that it was the ads that are eating up data. I was told wrong?
You were not told wrong necessarily. For sure the ads are consuming data. More accurately, everything you are displaying on your screen when browsing forums is consuming data. That includes things like avatar images, embedded images in posts, side bar links and general web page elements. All of those things (and more) are eating data every time you visit any web page. If you think simply blocking ads is going to result in very low data usage I believe you will be disappointed.
A program like Tapatalk is designed to take an information dense page, like most forums, and condense that information into more manageable bites. You can choose to not display all of the avatars and embedded images. The web page itself isn't displayed, just the thread text and headers. In fact, you can choose to only view threads you are subscribed to if you wish. Or only new threads, etc. All of this is made with mobile devices in mind and is much lighter on data consumption as well as much easier to read on a small screen.
Try using the tools available to you before (a) buying a new phone and (b) rooting and installing ad blockers to only address one minor area of data consumption.