First things first. If you vacumm the floors first, the dirt will be vacummed out the the bevels. Also, "mopping" is not allowed. What you are doing is spreading dirty water around and it is going in the low spots and open grain, discoloring the flooring and bevels. As I said., the proper way is to vacumm first then SLIGHTLY damp mop with a hardwood floor cleaner such as Bona. I think once you get accustomed to cleaning the floors properly, you will not be bothered by the bevels. To restore the floors now, vacumm well. Then clean the bevels using the Bona cleaner, a soft brisle brush (toothbrush maybe) and a shop vac to suck up loose dirt and moisture from the cleaner. Once the bevels are really clean, then clean the floors surface with the Bona cleaner. You may want to consider a recoat at this time with Bona Traffic floor finish. Hire a pro to do this. If all else fails, yes, you can have the floors sanded and it will reduse the size of the bevels but without proper cleaning, you will have the same problems again.
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