What’s the construction material?
During the 60’s due to baby booms, many prefabs were put up,cast in concrete.
Many prefabs have been condemned over the years and i suspect if your property isn’t visibly brick built, it is built of failing concrete.
However, settlement cracks appearing on buildings of that age suggest the possibility of ground heave, were the ground is drying out/getting very wet to either extreme. This could be caused by various factors.
Hard driveways stopping natural moisture getting into the ground around the foundations.
Removal of large trees but not their roots.
Trees or thick vegetation such as Laurels, sucking moisture out of the ground around the foundations.
What ever the cause, is needs a structual engineer to asses the damage and write report.
‘Tell tales’ will likely to be fitted to measure movement.
As others have said, get on the HA’s back about this till tell get it looked at by a professional. To me it appears they are just trying to avoid repair costs.
IMHO
HTH