From what you describe, I suspect you have already answered your question.
One possible fix – other than replacement – would be an underground drainage system in the yard. I say this because of the possibility that water is intruding from under the slab. If you can get that water away before it migrates under the slab you may solve the problem.
Another solution is to try and seal the seam between the foundation and the slab with a good polyurethane sealant. This will not work forever and will require maintenance, but if nothing else it will identify if water is getting in from the top of the slab.
Another possibility: Is there underground downspout piping running under the slab? If so, it may have failed and ruptured, causing the water from the roof to be pumped under the slab. Try disconnecting the downspout temporarily and run it through a pipe on the surface away from the home and see if the leak stops.