I highly suggest you take the assembly apart. This wobble you feel may only require a better adjustment of the nut. But, it may also mean that the bearings and races in the housing could be worn or damaged. You can get a good feel for how well this part of the bike was maintained by taking it apart and inspecting. If the bearings look good then you can clean, pack them up with grease, and reinstall. I almost guarantee they need grease anyway.
I like to tighten on the stand with the bars on, and only allow the wheel assembly enough slack that it just barely can fall to one side after placing it straight.