I GOT YOU FAM (main blog of @generic-sonic-fan here!)
The TL;DR: is going to be this post with my three rules for writing Omega.
But otherwise- Shadow and Omega can be similar in their dialogue sometimes but the #1 difference is that Omega is snarky and opinionated and loud about it!!!
Shadow would rather die than say anything that could present himself as annoying but Omega is a huge fan of insulting people no matter how childishly. He literally tells someone in Sonic Crossworlds to "CRY ABOUT IT" when they complain about him.
He is funny, on purpose! Aim to have less jokes about how he misses a social cue and more jokes about how he deliberately ignores them. Or is otherwise trolling. Keep in mind that he'll never be self-deprecating when he's trying to be funny but he is not immune to leveraging his status as a "killer robot" for the joke to land.
He is shouting everything he says. If he ever lowers his volume, it must be for a very serious and emotionally potent reason
If presented with serious emotions from another person, he's usually caught off-guard unless he has previous experience with the person (such as Shadow and Rouge). He'll usually try to deflect any difficult conversation topics back to literally anything else. Usually about combat, or Eggman robots, or weaponry, or even small talk if he has to.
But, Omega knows more than he lets on. While he likes to present himself as a straight-shooting "killer robot" he does know about more than just killing eggman robots. He has an opinion on absolutely everything, even the stuff that would make you double take because wait, he cares about that?
Finally, this guy is allergic to following rules or even other people's suggestions. He hates being told what to do even if it's in good faith from friends. Best case scenario if a friend tells him he should do something, he'll verbally deny it, then go do it while grumbling the whole time. Again, that's best case scenario if he likes the person.
Finally I also have this post which, while older, I still stand by.
And if you're really in the mood for some self-indulgent character analysis from me, this post is suuuuper long lol
Apologies for dumping on your post but Omega is my #1 favorite character of all time and I saw this post in his tag :)