Certainly use less solvent. That which is going through cant be doing the rear surface any good.
Since a straight solvent works by dissolving the styrene so that the two surfaces run together and weld, and wood is not soluble in normal styrene solvent, the only way that the workpiece can stick is by a lot of it dissolving and keying into the wood. Same for the cutting mat.
As I have yet to manage to glue styrene to my cutting pad with solvent, I have to wonder just what your solvent really is. I tend to use MEK which has absolutely no sticking ability whatever on anything other than styrene (quite capable of making a mess of plenty of plastics, though), and whatever the shop has in a tube as cement, which has only very marginal non-styrene sticking ability.