I have this problem too. My mount has a good routine to handle this with it's All Star Polar alignment routine. But I've never understood why you need to be at Polaris to align. Presumably it's due to the stars rotating, making it more difficult to get an alignment process complete within an acceptable time frame. But I always wonder why if you know precisely where the stars are, why you couldn't do a polar align from another star because you know it's relation to Polaris and the pole.
Some mounts have software to tell you how off your PA is after a 3-star alignment. Then you have to go through a correction procedure. This method works but is iterative and can take some time. I just did this with a friend on our last outing and I think we had to do 4-5 iterations and stopped because it seemed to always have a few seconds of residual error. On the last iteration, one axis was ~0 but the other was 2-3 secs. So it was almost as time consuming as drift align. I saw a white paper on this and its not as simple as it sounds. If I recall, there is an issue with the geometry of adjusting according to a star that is not at the pole. Also, you will have cone error in the mix and the Kstars method is not affected by cone error. In fact it works with the scope completely misaligned. I use my guide scope which is not always perfectly aligned.