I wonder what Dracula sites (plural) you have on the list, other than the castle in Bran.

For 4 days, this is what I would put on the list:
Day 1 - leave Bucharest and go to Sinaia (90min drive). Spend there 1/2 day at the Peles castle and then head for Bran (90min drive).
- spend the afternoon in Bran and spend the night in one of the many family homes or mini-hotels in the area. I can recommend "Vila Bran" or "Pensiunea Katerina", both very close to the castle. Enjoy traditional food there

Day 2 - head for Brasov to visit the city centre and the Black Church. In the early afternoon drive to Sighisoara, and spend the entire afternoon + evening in the old town there. I don't know many other interesting places in Brasov, so I would not spend more than 3 hours there. If you enjoy it though, spend the whole day in Brasov and head for Sighisoara in the late night.

Day 3 - half-day in Sighisoara and the late afternoon + evening in Biertan castle, ~30km away from Sighisoara. You can then find accommodation in Biertan, or head back to Sighisoara

Day 4 - Drive 60-80km to Sibiu. Great city to spend 1-2 days there, but half day should be OK to stroll through the city centre, Grand Square and surrounding areas. In the afternoon drive back to Bucharest, 4-5hr drive. Ideally you can plan in such a way that you fly back from Sibiu, they have daily flights to Vienna and Munich. For accommodation there, there's no other choice but the Hilton

You will need a car, as trains are very inefficient for this route. You can easily get one for 25-30 eur/day in Bucharest.