I would like to mix conducted tour and independent travel

Dear Jagdish, 10 days are too long for Beijing and Shanghai tour only. Actually, 6 or 7 days are enough to cover the main attractions of these two cities. Hereunder is a suggested itinerary for your reference:
Day 01: Tiananmen Square, Forbidden City, Temple of Heaven
Day 02: Badaling or Mutianyu Great Wall
Day 03: Summer Palace, old Hutong
Day 04: Take high-speed train to Shanghai. Xintiandi
Day 05: Shanghai Museum, Yuyuan Garden, the Bund
For the rest of the days, you may consider taking a side trip to Suzhou or Hangzhou or water towns around Shanghai.