It's a direct object, and "you" is the indirect object. Alternative construction, "I've tried to tell that I'm sorry to you" (with "you" as prepositional complement (in linguistic talk) rather than "indirect object."
The alternative construction is less likely (in my view). Usually, the direct object can appear immediately after the transitive verb; however, English likes to put "heavy/complex elements" (such as "that-clauses") at the end, which is why "I've tried to tell you (that) I'm sorry" is the preferred wording.
An adjunct is an element in sentence structure that can be omitted without altering the basic meaning of the sentence. Without any context, dropping "I'm sorry" (leaving behind "I've tried to tell you") won't allow us to properly understand the full intended meaning. Put it in context, and we can drop "I'm sorry:" I've tried to tell you I'm sorry. Yes. I've tried to tell you, but you won't talk to me.
No, I don't think it's a direct object. It's a complement of the verb, but it's what the Cambridge Grammar (CGEL) calls a content clause. It doesn't behave like an object in many ways. You can't use it as subject of the passive. You can't coordinate it with an object:
I'm sorry was tried to tell you.
I tried to tell you I'm sorry and the truth.
I tried to tell you the truth and I'm sorry.
A content clause is just another name for a noun clause. CGEL uses the term "content clause" in a specific sense: to differentiate this type of subordinate clause from relative and comparative clauses. Off the top of my head, I don't recall CGEL arguing that we should use the term "content clause" instead of "direct object" when discussing certain aspects of transitivity.
I'm using the term "direct object" for a good reason: to differentiate the direct object "(that) I love you" from the indirect object "you" in this ditransitive construction, where the direct object is closer to the idea of "core argument" of the transitive verb. The fact that "(that) I love you" can't pass the "passive" test doesn't mean that it isn't a direct object. Lots of transitive constructions don't have or accept passive transformation, due to semantic/pragmatic reasons.
And CGEL makes a distinction between "content clauses" that function as complements of verbs (i.e. "licensed" by verbs: He knows that I love you; I've tried to tell you I'm sorry) and those that provide information but are not "complements" of anything (verb, noun, etc.), as in You must be so proud of yourself, that he's been trying to tell you he's sorry.