Damp is an interesting issue and I'm no expert, just someone with my own damp to deal with!
There are many reasons for it and a variety of solutions....I have similar situation (but not very damp at the moment) and have read a book about damp and read up as much as I can online to understand damp better......
The general consensus on many forums is to be aware that many damp proof specialist companies will come out, probe for damp with the noisy flashing damp meter and then quote for an injected DPM and replastering.... while this may help in the short(er) term, it may not actually address the real source of the damp.
Some of the things to check for could include: blocked / insufficient airbricks, cavity wall full of old builders waste bridging the cavity gap, cavity wall insulation gone bad, concrete floor DPM allowing molsture between floor and wall edge, leaking pipes, poor ventilation, no extraction for bathroom and kitchens, people drying out cloths indoors, condensation on walls, bridged DPC, outside ground level about DPC, poor outside drainage away from house, porous bricks, pointing, leaking gutters down wall, splashback onto wall when heavily raining etc.
I'd suggest checking as much as you can before getting anyone in, but if you do get in a damp proof specialist, try and get one that doesn't sell a solution because they're obviously want to sell you a solution rather than you pay for their advice alone.
In my case, I've fitted a couple more airbricks and removed some debris from the cavity when the bricks were removed, I'm waiting to see if it helps when it next rains hard for a few days. I've also removed some plaster to see if the old plaster was causing issues etc.