The adjoint eigenspinor you multiplied by was the unit length eigenvector of $\sigma_z$ with positive eigenvalue.
If you you want a spin up result for the direction $(n_x,n_y,n_z)$ find a unit length eigenvector of $n_x\hat\sigma_x+n_y\hat\sigma_y+n_z\hat\sigma_z$ with positive eigenvalue. And use that instead.
If you wanted to do an interaction in the x direction and follow by an interaction in the z direction. Then you need to project onto the two eigenspaces for of $1\hat\sigma_x+0\hat\sigma_y+0\hat\sigma_z$ and then take each result and project them onto the two eigenspaces for of $0\hat\sigma_x+0\hat\sigma_y+1\hat\sigma_z.$ Where I wrote it in an overly complicated way so that you can do any type of directions, not just $\hat x$ and $\hat z.$
To be clear, if you choose the z basis (as you did) then the reason you multiplied by $[1,0]$ is because it was the adjoint of the eigenvector of $\sigma_z$ with positive eigenvalue. Do the exact same thing with $\sigma_x.$
If I don't know what physics concept you are asking about I can't explain the concept more clearly. Pick a direction, get a matrix, find an eigenvector, normalize it, take its adjoint, multiply by your vector, take the magnitude of the result, then square that. Done, that's the probability. Repeat for each eigenvector of the matrix.
If you are doing repeated measurements actually project onto the eigenspaces of the matrices. And take the square of the magnitudes of the projections.