I don't think you are doing anything wrong; I have read plenty of reports of the same behavior (not moving at all, or moving well past the 60º) while the ASIAIR performs polar alignment.
The maths right close the spherical coordinate's pole is very prone to numerical error. So much so that I always start my polar alignment with the OTA pointed a little away from the pole. Polar alignment has never failed me, and I wonder if my precaution saved me -- if so, perhaps it will work for you too.
May I suggest that the next time that you start polar alignment, do not point the OTA directly at the pole (declination +/- 90º). The polar alignment exercise is to get the polar axis itself pointed at the pole; the OTA does *not* need to point at the pole.
Indeed, according to ZWO's documentation, the OTA can be up to 30 degrees from the pole to perform polar alignment, although I have discovered that the alignment accuracy is off the further the declination is away from +/- 90 degrees by using my mount simulator.
I start at a declination of +89.5º before entering the Polar Alignment window. In your case, I suggest using the ASIAIR GOTO (or hand controller) to move the declination to between -87º and -89.5º before starting polar alignment. See if the problem goes away. But be sure to be near to the mount when starting polar alignment, in case the mount still runs amok.
As a matter of fact, I don't even start polar alignment at the "counterweight pointed at ground" position, i.e., with OTA at top dead center above the RA axis. Instead, I start with the OTA tilted about 60º (4 hours) east of top dead center. After the automatic 60º westward rotation, the OTA ends up at top dead center above the RA axis. The reason I do this is because my OTA train blocks me from tightening my RainbowAstro mount's altitude locking bolts when it is tilted by 60 degrees from top dead center. As a result, I want the ending position to be close to top dead center.
If you don't have such blockages, you won't need to do this. I only mention it here to point out that not only does the declination need not be +/- 90º, the ASIAIR also does not care what the starting and ending RA angles are, as long as they are about 60 degrees apart.
Good luck,
Chen