I have read about how some bakers have issues with getting the greasy stains off their silicone mats. There hasn't been a cure-all for this problem. The sticky texture mostly stays. I have using laundry detergent and scrubbing it in with a toothbrush, which did make it feel more smooth. However, the stains were not removed. It seems that oil conditions silicone to change color. I am at the point of just accepting that silicone just feels different after time has worked on it. I am looking for that soft smooth texture that I loved so much about new medical grade silicone, but through time my silicone just isn't that way. Maybe, I need to accept that this is a downside of using silicone and needs to be accepted. As it ages, it changes color and texture where oils have come into contact with it. People complain that dust and lint can accumulate on silicone. I just suggest that you air-dry silicone or even use a hair dryer. Water is the number one way this sticking occurs.
Another thing is that silicone is inert, in that it has a limited ability to react chemically. It's chemically inactive. So, silicone is completely safe for the body. Medical grade silicone is commonly used in the body.