Weepers (French drains) by themselves are not waterproofing. And the drywells are only the places where the water drains to and hopefully disperses without just coming right back where it came from.

Whether you can use inside or outside waterproofing depends on the foundation. If it's fieldstone or similar loose-laid material, exterior waterproofing is out. That's because the wall cannot be exposed from the exterior without being in serious danger of collapsing; in contrast to modern poured concrete or modular masonry construction, fieldstone foundations were built from the inside of the excavation. They were never meant to be freestanding.

Waterproofing from the inside is sometimes done, but many experts advise against that too, as inside systems tend to suck the earth from under the foundation.

Nineteenth century basements were never meant to be used as living space.

Here is a link that might be useful: Stone Foundations

In addition to the excellent advice from worthy:

It's more than a little possible that an 1880 house was built with an interior gravity drain system which has become plugged over time. Locating this and cleaning it out would help a great deal if one exists.

If there's a cleanout opening for the French drain system, that would be worth checking, too.

Exterior grading and the gutter/downspout system should be checked to insure water is draining away from the house.

There are interior systems which are designed to direct water to a sump where it can be pumped out. I have no direct experience with these, so can't say how well they work.

The fact that the basement was finished suggests that sometime in the past there was not the problem with water you're experiencing. That in turn suggests plugging of the drain system and/or erosion/changes in the exterior grade.

I have a feeling that the basement section that is finished was dugout at some point , as I think the house was all crawlspace at some point.

Whom do I call to make sure the existing system is clean, a plumber or a gardener?

Dug out or not, it doesn't seem reasonable that anyone would have gone to the trouble and expense of finishing any part of the basement if it was subject to frequent flooding.

First, I'd check for an existing interior cellar drain. Look for a place where the water accumulates to a greater depth than the rest of the area. It would also be the place where water remains longest. Look for some kind of opening out through which the water could drain.

Second, I'd see if you can find out who installed the french drain in 1998. May still be a map on file of the system.

As for who to call, look online or in the Yellow Pages under "drain cleaning". Some plumbers also do this, but not all. You need someone with a TV camera equipped snake and a high pressure water jetting system. Trouble is, you, or whoever you call has got to find an opening to the system, whatever it is, first. You've also got to hope that there is a repairable system. That's why finding out all you can about what's there is so important.

On a more hopeful note, if you've experienced a once-every-100-year's worth of rain this summer, the drainage system may just be temporarily overwhelmed.