Damp proof course advice.

If you have damp patches then as you say, you need to find the source before considering potential remediation. Inadequate ventilation; water tracking in from outside / the roof; a leak somewhere etc. Rising damp is somewhat lower on the list of likely causes.

Basically what Coug' said, look for sources of moisture, not always easy. So 1950's should be a cavity wall, look at the corresponding areas outside**, anywhere near down pipes or drains. Also 'damp' patches can be stains and salts in plaster attracting moisture. So first look at moisture sources, rising damp as such is rare, but leaks, faulty gutters, roofs etc are not, but the moisture has to get through the wall. Your initial phase includes 'appears' so yes on right lines more investigations, also check RH levels.
*** appreciate it could be an internal wall, chimney wall, any extensions or building alterations in the past could have triggered an issue too.
If it's a real concern (monitor for a while if you can) you can test the moisture for levels of minerals and see if that helps, but first check gutters, leaks etc. Hope that helps.

All the good advice is above.

It's rarely rising damp (but possible) and condensation is more common than most people think and usually confused with penetrating damp or rising damp. 🙄

Understanding the construction of the wall and footing and their present condition is pretty fundamental, otherwise you're just guessing.

My Mrs recently had this,

Expert said leaking drain tracking over 5m due to subsidence

Turn out to be air brick where slate covering it for years had been removed by tenant

personally I love an expert

Always have someone you know check

Thanks chaps.
The next step is to find a trustworthy surveyor.