Hi, M. The first thing that I would do is to install some towel bars. Damp towels sitting anywhere lead to all sorts of nasty things growing and even nastier smells eventually. The faster that you can get drying air to towels, the better. (Now if I could just convince our teenagers of this and that leaving wet towels in a pile on the floor is not a good idea.)
The second thing is to check your phone book for suppliers of cultured marble counter tops and visit several of them to see what they suggest. There are compounds for polishing cultured marble and they will have a better idea on how much work it is to do this.