Tokugawa period was 1603 to 1868, according to Wikipedia. That's like asking someone in the U.S. to see what influences there are of Pilgrims, settlers, pioneers, covered wagons, gold rush miners, railroad workers with slave labor, and general slavery still left behind.
You have time. Plan what exactly it is you hope to look for. Offhand, I'd say you won't see as much as you'd like, and that it's going to be more customs and traditions than sights. Tough call.
Does anyone know some places on Honshu with a strong connection to the Tokugawa period, or places where the modern culture is present.
The red part of your question makes no sense to me. This is 2014. What do you expect to find in Japan that isn't modern? We have indoor plumbing, electricity, skyscrapers, subways, electronics industry, pop music, international cuisine, zoos, museums, refrigerated foods, etc.
I also would urge you to look farther than just Honshu. You have a month, after all.
I suggest you focus on big things like footwear customs and clothing worn at shrines/temples, government, economy, etc.
Wikipedia says, "The individual had no legal rights in Tokugawa Japan. The family was the smallest legal entity, and the maintenance of family status and privileges was of great importance at all levels of society. " Not sure how you could examine that, but it's definitely something that has stayed with Japan. It even expanded to more of a sense of community, not just family. Perhaps the morning radio taiso or existence of sentos would be something along those lines?
It also says, "Economic development during the Tokugawa period included urbanization, increased shipping of commodities, a significant expansion of domestic and, initially, foreign commerce, and a diffusion of trade and handicraft industries. " Seek out Yokohama harbor. Look at the stock market. Take pictures of foreign products in supermarkets and on the streets (McDonalds, Fords, Coca-cola!).