i would think if you have to push so hard that you leave grooves in your mat you either need to replace your blades more often or definitly switch mats- my 24"x36" mat took almost 10 years before it was faded- kind of needing to be changed out- you really should not have to apply much pressure when cutting with a rotory cutter- and you should never use a knife or other tool on a cutting mat- they are made for rotory cutters- anything else will damage the mat- once there is signs of damage it is time to change it out- for safety sake.
i would cut it into smaller (good areas) mats to have other sizes & still be able to use the areas that have not worn out.- soaking a mat would ruin it- they are not supposed to be submerged in water- you can clean them with a soft cloth- and there is a tool (like a sanding disk with a handle) you can use to smooth worn mats and make them last a bit longer.