Positively connote and negatively connote sound wrong to me (native AE speaker).

Instead, I'd say has a positive connotation or has a negative connotation.

In your example, I'd say The author uses adjectives that have positive connotations.

To connotate as a verb also seems unusual. Looking it up, it appears to simply mean to connote, which is much more widely heard. So I'd say forget about connotate and just use connote.

You can say that a word connotes X, meaning that it implies X without explicitly stating it. What is connoted is a meaning. The connotation is the meaning that is implied but not stated explicitly.

Nonetheless, saying that a word is positively connoted or that it connotes positively sounds like, at best, a very awkward usage.

words have a positive/negative connotation is used here, for example.

Dann wird darin auch noch eine Wendung mit dem Adjektiv "connotative" verwendet, wie man sie gelegentlich im Netz findet: "positive connotative meaning".