The Museum sell silk products. You may go there to buy some. Or you can go to some time-honored shops, for example, Qian Tai Xiang (乾泰祥), which is located on Guanqian Street. The prices might be higher but the products are genuine.

Thanks for the advices.
We just came back from China.
In the end, we went to the silk musuem, New Silk Road shop, DongLing group silk shop, and shopping mall that sells a time-honored silk brand.
All 3 places have their own website.

The silk musuem is quite small. My father grumbled that we shouldn't have paid the RMB 15 entrance fee and waste time here.
But, we(the ladies) did enjoyed some photo-taking at the Silkworm-Rearing Room.
In the end, there is a shop. The products here are a bit dated & old(as in covered with dust & some were even torn & yellowish). We didn't buy anything here as I know we will be going to a much bigger silk shop(the dongling group one).

In the end, we bought the silk quilt in the New Silk Road shop because there was a 20% discount because we were brought there by a taxi driver(they have some kind of arrangement for bringing customers there, i think)...although I am not sure if we bought a genuine product or not.

New Silk Road and the DongLing group's retail prices are very similar.
The time-honored shop in the shopping mall near ShanTang Jie(in front of the water fountain near McD, Pizza Hut, etc) was cheaper by about 100rmb for the quilt i was looking at(also has a branch in GuanQian Street).

Hope this can help the next person.