I do think Acts was written c. 95 CE and I think its author(s) used Josephus, but it makes more sense to me to suppose that Josephus' account of Saul resembles what the NT says because he is the historical Paul.
By the way, Josephus went to Rome to argue for Jews imprisoned there and was shipwrecked around c 63 CE about the same time as Saul/Paul was sent as a prisoner to Rome in Acts.
Josephus' only interest in Rome was securing the release of Jewish priests as soon as possible and then returning home, though I suppose it's not out of the realm of possibility that he met Paul there too, particularly since I think his patron Epaphroditus could have been Paul's follower of the same name.
... I became acquainted with Aliturius, an actor of plays, and much beloved by Nero, but a Jew by birth; and through his interest became known to Poppea, Caesar's wife, and took care, as soon as possible, to entreat her to procure that the priests might be set at liberty. And when, besides this favor, I had obtained many presents from Poppea, I returned home again.
No person called Paul wrote a single letter in Acts of the Apostles.
It sounds to me like Paul wrote a letter to Ephesus in Acts 20:17, as I noted above. The word for "sent" can have the sense of writing a letter (https://biblehub.com/greek/3992.htm) and Acts doesn't say that Paul had sent people to Ephesus ("From Miletus, Paul sent to Ephesus for the elders of the church").
Saul got letters from the High Priest to arrest Jews in Damascus and also got letters from the Jerusalem Church. The contents of the supposed letters from the Jerusalem Church are also documented in Acts.
Acts of the Apostles is not a witness for the so-called Pauline Epistles so you have no NT corroboration at all than an apostle called Paul wrote Epistles to Churches
Acts shows that Christians wrote letters to each other, possibly including Paul to the church in Ephesus. Do you suppose Acts mentions every letter that Christians wrote to each other or has to mention the letters that happened to end up in the NT?
You seem to have forgotten that the author of the Epistles boasted that he spoke in tongues more than anyone else. The Pauline writers just don't make any sense. If talking in tongues was incoherent and doesn't build up the Church why does he boast about himself talking incoherently. And if believers cannot understand talking in tongues why would unbelievers?
Paul thanks God that he speaks in tongues more than other Christians because "he who speaks in a tongue does not speak to men, but to God" and thus thinks it has no benefit to other Christians without interpretation ("Now, brothers, if I come to you speaking in tongues, how will I benefit you, unless I bring you some revelation or knowledge or prophecy or teaching?"), but unbelievers were apparently more impressed with talking incoherently.