Yes sunspots do have proper motion. This usually depends on the type of sunspot and how active the sunspot group is. The best type of sunspots to determine the solar rotation rate are single penumbral sunspots that are circular in shape (when corrected for foreshortening). These sunspots are not particularly active and do not change appearance very much from day to day.
An example of large and small proper motion can be seen in Fig 24 of the paper ‘Sunspot groups without active region numbers’ in the June Journal.
At the moment we have a good range of sunspot latitudes, from about 15 deg to 30 deg N and S of the equator. Thus the sunspots at the lower end of this range will have very little differential rotation (the Carrington rotation rate corresponds to a latitude of about 18 deg) while the higher latitude sunspots will show much more differential rotation.