Hi Eshek
We had our son and family visit us here in Silistra this year. They looked into hiring a car in Bucharest but it worked out very expensive plus they retain a deposit of 900Euro becaue you have to cross the boarder. It was all very complicated. So we asked our taxi driver to pick them up he did this for 150lev he was there waiting for them, he took care of all the barge crossing and taxes etc. Now we used him all the time.
The journey from Bucharest to Dobrich is 4 hours drive
Constanta to Dobrich (1 hour 15 minutes)
Willow suggestion re a taxi then hire in Dobrich sound more of an options.
Where are you flying from?