A barn I used to work at just had dirt floors and no mats, when the horses would pee in them we would use this powder stuff called stall dry it works really good heres a link to one brand Stall Dry Deodorizer - 40 Pound
Something you could do is have some pea-sized gravel hauled to your barn lay down several inches of the gravel on top of the dirt floors then rebed the stalls. The gravel will help some in wicking the moisture away from the bedding. When you can manage getting rubber stall mats just put them on top of the gravel. When my DH got our barn built we put down gravel then bedded the stalls. It does help.
We put the gravel down on top of the dirt floors. There really is no need to remove the gravel when the mats are put down, just put the mats on top of the gravel. If we had went with the French Drain it would have been far better to do, though. A French Drain, as I recall, is a trench dug in the stall to about 3-4 feet deep then fill it with gravel to the level of the stall floor. Then bed the stall as usual. The moisture and urine will flow into the gravel-filled trench. Putting powdered lmne down on the floor of a stripped stall is a very good idea. It sweetens the smell of the urine.