Check you exterior drainage. A lot of older homes have drainage problems and the previous owner may not have put any perimeter drainage around the addition to the home, so water is likely coming in from outside. You may have to add exterior drainage yourself so that you don't end up with water in your crawl space. You might want to talk to someone that installs roof gutters because they may know more about drainage issues and be able to give you suggestions.

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Wood warps - cups - like this when the moisture is unequal. In other words, one surface, in this case the one facing the crawl space, remains damp while the upper surface dries. The solution is to install a vapor barrier over the surface of the crawl space to prevent moisture from reaching the flooring. The vapor barrier can be heavy poly sheeting, but any seams and the edges need to be carefully sealed. Ventilating the crawlspace will also help.

It's a little odd that none of the four contractors suggested this. You don't need to be an old house expert to know that a vapor barrier is needed in a crawl space. Your French drains may be working just fine. The moisture in the earth is generally always greater than that in the air, let alone the interior of a house.

A properly conditioned crawl space is the best solution. By all means, try the poly barrier first.

There are companies that specialize in retrofitting crawl spaces, especially in the humid south where vented crawls abound.

Here is a link that might be useful: Conditioned Crawl Space Construction