There are confusing number of types of dualism, i am not sure which one to restrict the question to.
Reductionist physicalism has a dichotomy between determinism and random. Dualists like to claim that dualism offers a way out of this dichotomy, e.g. with the mind being a cause without causes, but supposedly also not being random.
But just because mind is not bound by physics does not mean it is automatically not deterministic, or not random. Even if it was a causa sui, a cause upon itself, that does not mean it is not random. Even gods in paradise could be fully determined, or partially random, victims of whatever medium they are made of.
So what proof has ever been offered that any mental experience of my life was not deterministically derived from prior experience, or otherwise erupted randomly from nothing? If suddenly I feel an impulse to buy a hamburger, that motivation could have arisen deterministically, or by pure divine random. How could I subjectively exclude those two, and what is the third alternative in dualism?
"I want the hamburger because I want the hamburger" is just circular. I just want it means random, i might have wanted a pony instead. I want it because i am hungry and want satisfaction points to determinism, with the impulse being a causal effect.
Obviously i can experience my mind not being "single minded", i can refuse to give in to urges, but that's also possible with determinism, with the decision to not give in being determined.
What would a mental experience be like that we could logically identify as neither determined nor random?
Note this means specifically I am not interested in compatibilist accounts of deterministic dualism, I am interested in those arguments claiming dualism allows "non-deterministic free will".
Of course it can be claimed that simply all experiences are neither determined nor random, but that's just a claim.
One thing I could imagine is if i did not experience myself like in a movie shot from a point of view, but like the director in the cutting room seeing the whole movie start to finish and modifying it. But i don't experience my own existence that way, and i doubt anyone does.
For inspiration, the problem of introspection to declare oneself "free" are explored e.g. in publications like this: https://thorgan.faculty.arizona.edu/sites/thorgan.faculty.arizona.edu/files/Horgan_&_Timmons%20(2011)%20Introspection%20and%20the%20phenomenology%20of%20Free%20Will%20-%20problems%20and%20prospects.pdf
Are there any other such examples?