I’ve read a lot of third edition, but not been able to play it. But back in the late 1990s I played in a first edition Exalted chronicle as the only Dragon-Blooded character in a Circle of Solars. I’ve also adapted that character to the third edition rules to help myself get a handle on them.
The biggest differences that apply here are the totally revamped combat system, and the fact that players start with more inherent Exalt abilities and Charms - especially Solars when you include the free Excellencies. But from my reading, the differences between exalt types are comparable between editions: the Dragon-Blooded starts off with many advantages, and if you pay attention to this at character creation they’ll probably be the most powerful in their niche, at least at the start. This is especially true if the Dragon-Blooded character is a Dynast, and if they take advantage of those extra points to buy things like artefacts or supernatural martial arts. (Artefacts seem more powerful and expensive in third edition, so starting with one is a bigger expense that a Dragon-Blooded can more easily afford.)
This advantage fairly quickly fades as the characters gain experience, and the third edition Essence rules means that gap eventually grows much wider as they are slower to reach 4 or 5 Essence (and it’s more of a limitation for them as well). A Solar or Lunar with similar areas of expertise will easily outshine them before long, so you’ll want to make sure the Dragon-Blooded has particular talents that the others do not. My character was an Air-Aspected Magistrate of House Ledaal (a concept that doesn’t quite work in the revised setting of third edition), which meant he had good investigation, perception and knowledge of the Realm - not things covered by the other characters, who were warriors, mystics and sorcerers. He also had good family and government connections, and a position of authority that the Solars could never have; and started with some basic but magical artefact armour, a weapon and even a few hearthstones. (These were outshone by the orichalcum relics we eventually found for the others, but that was okay by me.)
And while Dragon-Blooded charms are less powerful, again there are areas in which they can do things Solars can’t. They have unique Aspect powers, including the Anima Flux which can damage enemies a little each turn. Their Charms have all kinds of cool elemental effects that Solars can’t emulate, like increasing the weight of an opponent’s weapon or armour (Falling Mountain Fang). And even though their Excellencies aren’t as easy to get or as powerful, some are more reliable or useful in certain situations; a Solar with only a dot or two in Sail might still fail a roll even with Salty Dog Method and a free Excellency, but a Dragon-Blooded with Fine Passage-Negotiating Style can buy automatic successes, and both re-roll 6s.
While most of these benefits are on the character sheet, there’s plenty else that a carefully chosen Dragon-Blooded character might bring to a Circle that means they can pull their weight, at least story-wise. In most of the Realm, and much of Creation besides depending on their background, they are afforded at least grudging respect. In most places Solars and Lunars are anathema to be destroyed at best, and dangerous unknown creatures at worst. Being able to hide behind a Dragon-Born will help them avoid suspicion, and also open doors that would otherwise be closed to them. In my game, the Solars posed as Archons working with my Magistrate, and this worked well enough for a while that we could travel pretty freely.
In modern Creation there are places where the opposite is true - Lunar domains and cities where Exigents or other Exalt types are revered generally are no fans of Dynasts, for example.
Really, the most important thing is to talk to your players and make sure they get it. If you’re mixing Solars and Dragon-Blooded then you ought not to prioritise mechanical balance in the sense of everyone having equal power according to the dice. Instead, get on the same page about the story you want to tell, make characters who have a reason to need each other and work well together, and embrace it. Characters are not meticulously balanced in ability in fiction, and they needn’t be in your game for it to work either.
Story-wise my character needed some convincing that the Solars weren’t dangerous anathema, but once he was he had to reconcile this knowledge with what the Realm had taught him. He felt he could do good steering their actions, and they fought (and defeated!) a Deathlord together. But the story also became about his family worrying if he had gone astray, and trying to work out what to do about the conflict of his duty to them and to the greater good. All of this worked in well as an extra level of drama and complication to the choices the Solars had to make, as well, even if after several sessions the Solars could all wipe the floor with enemies I couldn’t touch.