If you observe Uranus and its moons orbiting our sun, the trails will make "spiral" patterns. However, if you wait for Uranus to move 90 degrees more around the Sun, the moons' orbital planes will no longer be perpendicular to the planet's direction of travel. Instead of moving around Uranus's trail, the trails of the moons will slice through it vertically.
Is there any way to correct this, so that the moons will make spiral trails for the entire orbit? Is this even possible in real life?
To answer your first question: not that I know of; for the orbits to always be spiral-shaped, the orbits would have to precess at a constant rate (Orbital precession is best described as a change in Longitude of Ascending Node). As far as I know, there is not a way to do this in US2.
As for your second question: Some satellites have orbits like this:
http://en.wikipedia.org/wiki/Sun-synchronous_orbitWhat you are describing is equivalent to a Sun-synchronous orbit with the Ascending Node twisted 90 degrees.
However, these orbits have very low altitudes- on Earth, they lie within the LEO range. For most planets, this would be well inside the Roche limit. Although spacecraft and very small moons could have such orbits, large moons would break up.
One final note: the longer the parent planet's orbit, the farther out a sun-synchronous or spiral orbit can be. I don't know the exact math, but you could probably calculate how far out a planet would have to orbit for sun-synchronous orbits outside the Roche limit to be possible.