Welcome to Reddit u/Proper-Plenty4687, sorry for the late reply.

We don't have a lot of detail about the administrations of the rivals to those that would later become the "three kingdoms" but most claimed legitimacy via the Han. Via being appointed by the Han controller, later confirmation in position or, if not granted such official blessing, portraying themself fighting against the controller on behalf of the Emperor. As a proclaimed loyal servant of the Han drawing legitimacy upon it, they mostly (the theocratic Zhang Lu went with a different regime with his five pecks of rice sect) drew upon the existing system of Han administration.

If a warlord controlled multiple provinces, the one that wasn't their base would be put under an Inspector or (thanks to the revival of the rank in 188), the more powerful role of Governor. An inspector would supervise and keep an eye on things, and a governor had more direct control but both could raise troops across the province and they were meant to be from outside the province to help counter local influence. In Yuan Shao's case, he gave such roles to two of his sons and a nephew but most warlords relied on either officials they could trust or acknowledged the situation on the ground and appointed a locally powerful figure (like Zang Ba being given charge of Xu by Cao Cao on the fall of Lu Bu) who knew the area and could bring local support.

Commandries, including the major cities, were managed by a Grand Administrator (unless a King's fief, in which case they were a Chancellor). This could be, when things were relatively normal, someone from outside the province coming through the ranks but areas of importance, whether due to stability or key military position that needed securing, might be put under a relative or a trusted official. For example Shi Xie's use of relatives to assert the family control across their native Jiao. Or the official Man Chong being sent to the Yuan traditional home area of Runan to destroy Yuan support. They might also hold the title of general which gave them authority to gather troops beyond the commandry to help defend their area. Sometimes it would be rewarding a local supporter with power and prestige in their locality or recognizing and confirming the authority of a local powerful figure in exchange for continued loyalty.

Governors/inspectors and administrators would have their own staff, meant to be drawn from the local families though they might also hire refugee scholars into their service. Below the commanderies were circuits which were overseen by magistrates whose circuit size could vary but an average magistrate could expect 10,000 households (40-50,000) people in their circuit and often had more than that. It was possible for an official, with enough years, to build support (and a potential power base) but you might be able to imagine the limitations of a magistrate in keeping a close eye on all those households. The magistrates also had the issue of working out what their strategy was for the local magnates? To be strict and take them on with the violence that might require (or getting sacked for stirring up trouble) or to turn blind eyes and keep quiet.

The Later Han had problems with local powers during times of peace. The later decades saw tensions between the Emperors and allies like the eunuchs vs the powerful families, both at court and at the local level, tensions that could get violent. Officials were often from, or clients of, powerful families and shared interests with the figures they were meant to be keeping in check, relying on the local families not to cause them too much trouble for their career or their life.

The powerful families had patronage in the recruitment system and provided the officials that were needed to fill the bureaucracy, bought up wealth including the farms with many farmers losing their livelihood, and built up and armed their own retainers which sometimes went as far as being used to carry out murder, able to use the tax system so they and their clients didn't pay their full share leaving the Han struggling for funds to meet various crises and unable to always impose laws.

The civil war did provide some opportunities for governments under various warlords to try to strengthen control. Killings of local leaders were not an unknown method for new warlords (Liu Biao, Gongsun Du among others) to impose control, agricultural colonies (most famously used by Cao Cao) got (some) farmers directly under the control of the government where they could be taxed and used, using refugees and outsiders as a tool to balance against the local powers, careful use of.

However, one story of the era was the continued loss of control. Often warlords, having relied on people who brought their own men and resources or acted as key supporters to ensure local support, warlords then turned blind eyes to murder from figures that were still felt to be needed. There was a sense of the law needed to be applied yes, after the weaknesses of the Han but just maybe not quite fully just yet, not all there was unification.

Some warlords, like Liu Zhang, always struggled to gain central control over their powerful supporters Liu Zhang had relied on their support to succeed his father and struggled to impose himself. We don't get a lot about his regime but the tension between him and the powerful figures is a running theme. While a shift of strategy like Liu Biao's shift from northern to southern strategy had to be weighed with the cost of support from the northern families. Wu didn't just have inherited fiefs but inherited positions and troops, Zhuge Liang's relatively successful pacification of Nanzhong involved Nanzhong magnates having local positions by right with limited oversight, the Cao's accepting the Qingzhou troops hereditary positions.

When Chen Qun's Nine Rank System was intended to get recruitment out of the influence of local powers but the rectifiers were from the powerful families and ensured control remain in their hands. When Xiahou Xuan, a leading figure in the regency of Cao Shuang, attempted reform he met with opposition and the sense of the regency attempting to centralize control was one of the reasons for Sima Yi's coup. The internal change from the Cao to the Sima, or Wei to Jin, was at least partly replacing a family that was seen to not be acting in the interests of the gentry for one that would. Like not meddling with the Nine Rank system that allowed such control to the powerful families.

Even when the land was united, the records from the fallen kingdoms and Jin being unable to do a full census but relying on who they could tax shows that all sides had lost the ability to get access to the manpower and resources of their subjects. They were only able to access parts of the manpower and wealth of their lands while, during the civil war, attempts to build power bases or to tilt the balance their way could get a strong response from the powerful families to protect their own interests.

In terms of the average peasant farmer, we don't get many accounts from them. When dealing with officials, it would be the local ones and those supervising them for the most part and so changes in the personal and attempts to reign in the powerful families might have got noticed but the Shu-Han regime had only limited control in the area. A farmer might find the need to repopulate Hanzhong and develop the agriculture projects there useful, particularly if they had been forced out of their farms by the richer neighbours. But often the farmer might just be one of the 40-50 thousand people for a magistrate to handle and that is if said farmer was even on the tax registers rather than under the control of a local magnate.