What you're doing is assuming that we're physical things. But that's just a dogma of the modern age.
It is a self-evident truth of reason - one you yourself appeal to - that free will is incompatible with our decisions having been antecedently determined.
But it is even more powerfully self-evident that we have free will.
Consider: in the lengthy debate over free will the majority of those who have thought long and hard about the issue still conclude that we do, in fact, have free will. They either conclude that compatibilism is true, or they conclude that incompatbilism is true but that the indeterminism is says is necessary for free will actually obtains (libertarianism). Only a minority (albeit quite a significant minority) conclude that we lack free will. So evidently most of those who think long and hard about this issue recognise that it is powerfully self-evident to reason that we possess free will.
So this premise:
1. We have free will
is exceptionally powerfully supported by our rational intuitions.
Less powerfully supported, by powerfully supported nevertheless, is this premise:
2. Free will is incompatible with everything we do being antecedently determined (that is, if we have free will, then not everything we do is antecedently determined)
The conclusion that follows from these two premises is this:
3. Therefore, not everything we do is antecedently determined.
But as you point out, this premise is also true:
4. If we are physical things, then everything we do is antecedently determined
And what follows from 4 and 5 is this:
5. Therefore we are not physical things.