Now for the bad news. Although moving the Ring so the sun is out of plane leads to a nice gravitational restoring force, moving the Ring so the sun is off center doesn't. This can be demonstrated without calculating the whole force on the Ring, just by calculating the forces due to gravity from small arcs of the Ring. On the near side, the arc contains a mass proportional to (R-x) and the effect of this mass is divided by a distance of (R-x) squared. On the far side, the opposite arc has mass proportional to R+x and the effect is divided by (R+x) squared. These two effects don't balance - any slight offcentering will get worse and worse. The fact that the Ring is rotating doesn't make anything worse or better - I think (someday I will do a full-up analysis to make sure though).
Note that a Dyson sphere is much better off. The mass contained in an arc at distance (R-x) is proportional to (R-x) squared, so when this is divided by (R-x) squared to get a force, that exactly balances the force due to the opposite arc. The sun can be placed anywhere inside the sphere and will feel no force moving it elsewhere. In Ringworld Engineers Larry Niven revealed that the Ringworld has attitude jets along its edge which convert solar wind into thrust, making the Ringworld stable by providing a restoring force that increases properly as the off-centering gets bigger. The only problem is that someone has stolen most of the jets.
Let's talk a little more about stability, since I mentioned it.