Like any state, their priority was first survival then prosperity, ie. gaining relatively more power when they could. Political ideals were, most of the times, tertiary, and have to be read along with other goals, constraints, and opportunities in mind.

Let's tackle Wei first, since it will probably illuminate the issues better. It presented itself as the legitimate successor of the Han Dynasty, and to do so it had to defeat the remaining challengers to its sovereignty. However, it wasn't that clear cut. Throughout its existent, its wars with Shu were mostly defensive and there was only a few counter offensive attempts. Its attention was with Wu most of the time, despite the fact that, unlike Shu, Wu didn't attempt to directly contest Wei's claim of sovereignty: Sun Quan crowning himself as An emperor was a response to the pervious Wei's status as his overlord, especially about the issue of sending his son as hostage to Wei's court.

Superficially, it would seem that Cao Pi of Wei was willing to accept some type of peaceful coexistent or Wu's de facto autonomy for the time being, especially when he had just succeeded his father and had internal issues to deal with. He also rejected a proposal to attack Shu in the mist of Shu-Wu war. It would seem, if the hostage issue could be resolved, the peace could have lasted indefinitely.

That said, however, the peace between Wei and Wu wouldn't last in geopolitical terms, since they had a standoff between the Yangtze with each of them controlling one bank up stream, and the land North and South of the river was suitable for large scale military operations, and that's the reason why Wei's attention was on Wu most of the time.

The thing that drew that attention away was Shu's repeated offensives. Traditional history ususally described them as attempts to destroy the Wei, or "Conquering the North", but if we look at the military operations and their targets, they seemed to be something more mundane.

Shu's expeditions all targeted the West most areas of China at the time, modern day Gansu or as it called back then, "the Corridor West of Yellow River". It never had the economy of force or the logistics to go beyond the Gate of Tong and Chang'an. They were also launched after Wei had reestablished contact with the Silk Road, and accepted tributes from the Central Asian states. Suppose Shu was successful in its expeditions, the immediate impact would be cutting of Wei's trade route and establishing itself as the lord of it. Even Shu's Southern expedition had an economic reason behind it: with the conquest it gained trade access towards India and South East Asia. The Silk Road was exponentially more important, not only it would bring in a huge amount of wealth, it would also give them access to Central Asian horses, of which Shu severely lacked. And as we know, trading for horses was so important that Sun Quan was trying to established an overseas trade route with and backed the Gongsun regime in Liaodong.

Then, things sort of "cooled" and scaledowned for a while when the Sima had taken over Wei, and again their primary concerns were internal challengers and rebellions, and Shu had exhausted itself with its expeditions. The real conquest level expeditions only happened when the Sima had firmly established themselves and got tired of Shu's attacks. Wu was more like an afterthought and they weren't sure whether they could or should completely conquer it.

So, did they all want to unify China? Wei and Shu would say yes, but their actions could be better explained by the prospects of immediate gains; these gains could build up sufficient advantages that would allow them to fulfill their stated ideals in long term. Wu was a bit different, that is not to say it didn't "want" to unify China, it was just the most practical of the three: it would try to gain power when it could.

Did Shu give up restoring the Han Dynasty. If you judge by their actions, it kind of did after Zhuge Liang's death, and it was necessary because it was weak and exhausted. When Jiang Wei was in chargeb of its military operations, he was only allowed to launch small scale expeditions trying to take small pieces of territory. He would later be allowed to launch large expeditions, but they could never match the scale of those of Zhuge Liang's.

The three kingdoms were never fully at peace but there were some "peaceful" periods, when they didn't expect the others would be on the offensive. The more interesting question, however, would be whether they could maintain peace. Suppose Shu didn't bother attacking and Wu accepted its vassal status, what would happen then? This question could be extended to the North-South Dynasties and Southern Song periods. And the Southern Song kind of gave us an answer, as it could coexist with Jin after reaching an agreement.

This is my view by applying geopolitics and political-economy in my reading of the Records of the Three Kingdoms.