You use two different coordinates, polar and rectangular, without specifying how they are related. The relation I will assume is the textbook one, where
$$ z = r\cos\theta$$ $$ x = r\sin\theta\cos\phi$$ $$ y = r \sin\theta\sin\phi$$
So that the vector field
$$ v_x(r) = r \cos\theta$$ $$ v_y(r) = r \sin\theta\cos\phi $$ $$ v_z(r) = r \sin\theta\sin\phi $$
is radially outward. Then you flip the sign on v_x, so that the field is radially outward in the y-z plane, but inward in x. Another way to describe this field is:
$$ v_z = {z\over r} $$ $$ v_x = -{x\over r} $$ $$ v_y = {y\over r} $$
this thing sucks the flow inward to the y-z plane, it is a saddle flow. What you want to do is
$$ v_z = {z\over r} $$ $$ v_x = -{y\over r} $$ $$ v_y = {x\over r}$$
In other words, you need to flip x and y. So your convention for coordinates flips the normal ones:
$$ z = r\cos\theta$$ $$ x = r\sin\theta\sin\phi$$ $$ y = r \sin\theta\cos\phi$$
(this is different from the previous thing). In this situation, the gas will precess around the axis as you intend, the net speed will be constant, and the trajectories will be spirals around the z axis.