I like the french drain, too, and I don't see why it would need to be inside. I had a similar situation, with one major difference. This was a house in California, which like most there was built on a slab and had no basement. It had a hill in the back that sloped toward the house.
When we bought the house, the sellers disclosed drainage issues that had led to a lawsuit. They sued the owners who had sold to them for not disclosing the problems. When the people who sold to me lived there, they experienced wet floors in some rooms in the winter. There also was quite a bit of efflorescence on the garage floor and driveway, causing the surface to be crumbly. The diagnosis was that water seeping through clay from the next-door neighbor's house, uphill, and from that hill in the back would build up pressure in the winter and push through the concrete, making the floors wet.
The previous owners settled their lawsuit and built a french drain on the side of the house, where the hill was closest. We hired an engineer who recommended extending the drain around the back of the house, and we took that advice. We hired a company to come in with equipment to tear out the back patio and sidewalk, which were pretty ugly anyway. That company dug a trench for the drain to go out to the street. I built the drain, with the perforated pipe at the bottom and a solid pipe for downspout water higher up. Then I rented a Bobcat to regrade the back yard and added a short retaining wall at the base of the hill. Then came a new sprinkler system, a deck just off the ground, etc.
It was a big deal, but it worked! In the 17 years of owning the house, we never had any problem with wet floors or other drainage issues. So I'm a big believer in french drains.