funny dampness, If there is a interior drain pipe that's usually where it is.
wondering if there are window well drains connected to the perimeter drain that might be filling into the perimeter drain.
All this speculation re a drain , I doubt there is one unless there is a sump pit or your on a grade that would allow for daylight drainage.
I would drill a hole and look under the slab .
If there is water I would just put a sump pump in and be done.
Obviously it is a nicer house, and this is odd weather but fixing the symptom in this case without a major disruption is in order.

I would not call a waterproofing company under any condition. The slab might be on gravel.. very common these days and the gravel will allow water to drain to a sump pit hole.. The 10K french drain route is a classic ripoff. Seen too many folks get the treatment /shaft when it was obvious negative backpitch of the grade was the first option. Waterproofing companies around here are major thieves. What they do will work , but why anybody spends that amount of money to collect the water and not fix the problem is a sin. I have never seen a waterproofing job that I honestly felt the homeowner did not get riped off.