Cold War Brinkmanship/Diplomacy
Your dragons are relatively powerful, as mentioned by the other answers, so they're certainly stronger than a human army on its own, but there is a bit of a problem.
Females have a natural lifespan of around a millenium while males live ~500 years. Males are also more common, since females lay clutches of ~6 male eggs every few decades but only 1 female egg approximately every century.
That indicates an entire possibility that a Dragon Queen is a major target for any other Dragon Queen, since if you defeat them, either by subterfuge or by defeating them in honorable combat, it wouldn't take long for one Dragon Tribe to eventually find themselves effectively annulled by means of not having a female dragon who can lay another female egg within a century. Best case scenario, you end up with a comedic story about a Dragon Princess not acting like she was next in line to ascend to the Dragon Queen/Empress role, but that also gives large amounts of power to intermediate regents as a result.
In a situation like that, the dragons will likely advance up to and through Peace of Westphalia style treaties just to avoid further bloodshed, and likely leading to a League of Nations/United Nations approach of resolving conflicts by diplomatic means well before they get to a point where they actually get to a warring state.
All factions would be agreeing to this (Well, except possibly the human slaves themselves, but they would likely be held in line by the dragons' ability to enforce said peace) due to the sheer fear of what would happen if they were invaded by a different <PERSON>.
This said, it's not like they wouldn't posture for war
So, let's say that Nation X sends a messenger to the Dragon UN to try and claim more land from Nation Y as part of a dispute over resources on some bordering island I.
As they're doing that, they may send some of their male dragons to the island I, perhaps in pairs of 2, carrying entire makeshift crates of human soldiers, armed up with equipment to both fight enemy dragons, and enemy humans.
Then they just wait there and effectively siege the island (Or if there's a non-island territory being disputed, just surround the territory that is being disputed) while they wait to receive an update from the Dragon UN messenger as to the results of the dispute, and either occupy the island, or leave it, if it's determined during the treaty that Nation Y responded by effectively doing the same with equal or more force at island J, and they managed to get a concession out of the original invader, evacuate and leave.
If Nation Y doesn't necessarily want to occupy island J instead, and go to island I to confront Nation X, they would likely still be mostly about declaring stakes, and essentially boasting about the size and strength of an army they brought to the area to act as "muscle" for the negotiations on the treaty side.
This would likely be followed by a lot of public relations by both nations as to the motives of the opposing nation, and as to why they deserve the territory.
Ultimately, this would have a much larger effect on which dragon tribes have more territory based on which tribes have the better luck at having more spare Queens lined up in case of an disagreement leading to an actual fight, or how many male dragons a tribe can spare for these brinkmanship fights along with their human slaves to justify the other side capitulating to their demands - but full-on skirmishes, and outright wars, would likely be very rare, and would only come due to a large portion of diplomatic means breaking down, combined with dragon nations that are willing to risk lower level female dragons due to having enough female dragons to safely have a Dragon Queen buffer in case of a succession crisis due to the conflict getting beyond regularly heated. That said, even in those cases, <PERSON> might be trying to secure exit strategies should the battle turn against their favor quickly on a field of battle - because once the fight begins, the other side might just press their advantage to prevent the attacking nation from attacking them again in the future. 164