What does もっとたゆたゆver. mean in AT-X shows' broadcast information?
"Tayu-tayu" is an onomatopoeia word. Characteristics of a Japanese onomatopoeia include a two-syllable word in a repeated fashion. Many, if not all of these translated poorly into English. For an example fuwa-fuwa can be used to the fluffy texture of something (tangible or intangible, like a mood or figurative characteristic). They tend to be more sporadic, with little to no clear standard of usage (so they tend to be quite informal, generally), used a bit like pewpew, kaboom, pow, in a loose sense.

"Tayu-tayu" despite being the title of a popularly known explicit hentai anime and manga, it is in no way related to that series. Typically, it can be used to describe the swaying or shaking motion. It's likely derived from "tayuta", which can refer to something tangible like water, or something intangible like anxiety (like fluttering of the heart/mind). As with many things in Japanese, the context is very important to the actual meaning. In this case the AT-X version, is likely referring to the more "uncensored" nature of the broadcast, as for what is doing the "swaying"... you'll have to watch to find out.