I read in Kitab al-Ghaybah of Shaykh at-Tusi that the handwriting the letters of Imam az-Zaman were written in was the same as "during the life of Imam Askari SA".
Some Sunni brought this and claimed that Muhammad Ibn Uthman was fabricating letters during the time of Imam Askari up until his own death, and that Uthman al-’Amri was in on it. I have no way to refute this, and I’ll admit, if one didn’t know of the hadiths praising them and appointing them, they’d probably come to the same conclusion.

Are there any ahadith/eyewitness reports that the handwriting stayed the same through the time of al-Hussayn Ibn Ruh and ’Ali as-Samuri?

I really need an answer, to be put my own heart at ease.

So Uthman al-Amri was a companion of Imam al-Askari (a), and in Sahih hadiths Imam al-Askari (a) said that he is reliable and trusted. An infallible Imam here is testifying that he is truthful, he is to be obeyed and he is to be trusted. So if he later says these letters are from Imam Mahdi (a), we accept. Very simple.