Can the word "that" modify a verb?

Solution 1:

“that the indictment had been dismissed” and “that the court lacked jurisdiction over the case” are subordinate clauses. They are not adverbial: these clauses act as complements of the verbs allege and argue respectively. The relationship is similar to that between a verb and a direct object (a direct object is a noun phrase that acts directly as the complement of a verb). Compare the use of allege and argue as transitive verbs with a pronoun like what as the complement, as in “what they alleged” and “what they argued”.

Solution 2:

The defendant alleged [that the indictment had been dismissed] and argued [that the court lacked jurisdiction over the case].

In modern grammar "that" belongs to the word class (part of speech) subordinator, primarily because its main job is to introduce subordinate clauses. Its function in the clause is 'marker'.

The two bracketed elements are called content clauses, and their function is not object, but complement of the verbs "alleged" and "argued".