There is nothing wrong with the TW34. In dirt, the problem is the TW32. 50 dollars solves that.
ATV tires involve a whole 'nuther level of commitment. They're heavy, costly, tall and will slow you down on pavement, difficult (even dangerous) to mount on the rim, they are virtually impossible to remove from the rim in the field, and for greatest joy they often require other unforseen modifications such as chainguard removal, extended swingarms, etc
That's a pretty tall fence for many. Make sure you can utilize their upside frequently enough to justify tolerating their downsides. Their main upside is that they offer 110% traction pretty much at ALL times.
It may have just been a typo on your part, but you don't want anywhere near a 28 inch ATV tire cuz it won't fit. Stick to the 26X8 or 26x9 designators. All of these tires differ greatly in their listed size designator vs. their ACTUAL mounted-and-inflated height. Some of them are well under 26", others over 27", actual. An inch matters hugely in a stock swingarm. Shorter is better.
I started out with the plan of running two rear rims, one with the stocker, one with the ATV tire. In reality unless the trip is going to be over 500 miles I pretty much run it full time and live with the downsides because my riding is so mixed that ultimately it's just too much hassle. A more disciplined (less lazy?) person could employ the two-wheel system and enjoy their choice of the best of both worlds.