Thanking adverbs are provided in this article. The words listed here are commonly found along with the verb thanking in sentences. This reference page helps answer the question what are some adverbs that describe or modify the verb THANKING.
cordially, devoutly
heartily, humbly
publicly, silently
warmly
Hope this word list had the thanking adverb you were looking for. In addition to adverbs used with thanking, other example verb adverb pairings can be found on this site.
You can help by adding more adverbs in the comments. Thanks.