Although (unlike some people) I looked up the Complete Divine version of the Dragon Breath spell before posting, I just double-checked; there is no limitation on the number of such spells which can be in effect at any time. So a level 10 sorcerer could cast three of them then go into battle and have 7 rounds of overlap -- more than enough time to finish most battles at that level.
With three Dragon Breath spells running, the sorcerer gets at least three rounds of breath weapon attacks, and has pretty good odds of getting another attack on each of the next four rounds (depending on the 1d4 rolls).
Of course, there are better level 5 spells for a sorcerer, but for an NPC who's really into his draconic heritage, this would be a fun schtick.