I don't think that the て-form and the continuative form verbs qualify as an "adverbial" form, but at least they add meaning to the following verb just like adverbs.
流れている (流れる, an ichidan verb, in the て-form)
ご飯を食べて寝る* (食べる, an ichidan verb, in the て-form)
*everything preceding the て-form verb is included in the "adverbial" clause
ゆっくり話す (just an adverb for comparison)
決して忘れない (This is an interesting one. It is an actual adverb, but clearly based on the verb 決する.)
抱きしめる* (抱く, a godan verb, in the continuative form)
*maybe it is the other way around here, しめる adding meaning to 抱く
早く寝る (早い, an i-adjective from your example)
It should be noted that these verb forms have many other uses in addition to modifying following verbs. For example, the continuative form is used create the て-form:
godan: 抱い(alternative cont.)+て)
It attaches to a noun to create another kind of meaning:
The て-form can also be used in various ways that don't resemble adverbs in my opinion.
As Igor Skochinsky pointed out in comments, you can actually add -的 to nouns to make them adjectival nouns and as you said in the question, it will become an adverb when you add -に. The resulting adverbs aren't guaranteed to make sense, but that wasn't a requirement. Other ways to use nouns as adverbs depend on the noun in question. Some examples below:
Temporal nouns like 今年 can be both adverbs and nouns without adding anything, and nouns like 後 can be turned into adverbs by adding に or で in addition to being an adverb on their own in some situations.
Any noun that works as a verb with する can be used with the て-form (して) like ichidan and godan verbs in the examples. Like the examples, this isn't an adverb but something resembling it.