I'll try to answer your question, sorry if it doesn't satisfy you

This is a bit tricky since the Crimean Khanate was indeed an Ottoman 'vassal', yet they enjoyed autonomy or even independence in their domestic and foreign policies, as long it doesn't interfere with Ottoman affairs. The Crimean dependency on the Ottoman Empire "was in a continual flux between the late 15th century and the end of the 18th century". Let's tackle this separately

First, the Crimean Khanate controlled its domestic policies independently. The Khanate for example enjoyed the right to collect taxes and there is no evidence of any taxes from the Khanate being paid into the Ottoman treasury. The Khanate also had a judicial and punishment system that was a bit different compared to the Ottoman Empire, and that it was the Khan, and not the Ottoman Sultan, who was responsible for delivering justice to the Khanate’s inhabitants. The Crimean Khan therefore exercised the privilege of an independent Islamic sovereign to enact justice in his territory. Yet this doesn't mean the Ottomans were completely out of the picture. The Ottomans controlled a crucial and important part of Crimean Khanate's domestic affairs: its succession. So the eldest member of the Genghisid Giray dynasty was expected to be put in the throne, however a new Khan is chosen by kurultay, an assembly where nobles elected the new Khan. This, combined with the Ottoman understanding that they have the right to appoint a new ruler makes it so that the Ottomans could control who succeeded the throne. The political and personal ambitions of Crimean nobles and Crimean princes could be easily manipulated by the Ottomans who could turn the dispute to their favour. One such example is during the 1608 Crimean succession between Tohtamış Giray and Selamet Giray. The kurultay elected Tohtamış Giray to be the new Khan, but the Ottomans disapproved him and supported Selamet Giray. After the nomination, Selamet Giray left for the Crimea. At the same time, Tohtamış Giray went to Istanbul to seek the sultan’s support personally, but he was murdered by clients loyal to Selamet Giray in the vicinity of Akkerman. Afterward, the Ottoman nominee Selamet Giray seized the Crimean throne. Of course, there were times when a Crimean khan ruled without Ottoman approval, like Mehmed III Giray who managed to keep the Crimean throne thanks to domestic support and the backing of the Cossacks but most of the time the Ottomans controlled (or at least partially) who ruled and their influence grow stronger as centuries goes, by the 18th century the kurultay was mostly ceremonial as the Ottomans appointed Khans as they saw fit, though the kurultay could still force the Ottomans to depose an unpopular Khan.

The Crimean Khanate also exercised a lot of autonomy in their foreign affairs, even conflicting Ottoman interest at times. The biggest example of this was the Ottoman-Crimean conflict in North Caucasus. From the 15th century, the North Caucasus was effectively ruled by the Crimean Khanate. The Crimean Khans, via a system of tribute and installing Crimean princes as scion of Caucasian tribes nominally controlled North Caucasus, with the Ottoman Empire allowing it to keep the flow of goods and slaves to Ottoman territory. By the mid-16th century, the newly emerging Muscovite Russia threaten Crimean hegemony in the region. The Crimean Khanate fight the Muscovites and their allies, but Khan Devlet Giray was aware that without Ottoman troops and cannons, he couldn't continue the fight against the Muscovites. While he needed and welcomed Ottoman help for his struggle against the Muscovy, he was careful not to allow the Ottomans to extend its sphere of influence and control beyond the coastal areas of the northern Black Sea. At this time, the Ottomans received pleas for help from Muslim inhabitants of the Volga under Muscovy. So an Ottoman army under Kaşim Pasha was sent to conquer Muscovite fortress of Astrakhan and turn it into an Ottoman province that could act as a springboard for further expansion and a stronghold against Ottoman enemies, especially Safavid Persia. Devlet Giray realized the danger of this, as this would left the Ottomans completely encircled his Khanate. He then spend a lot of effort sabotaging Ottoman operations in the region without offending the Ottomans as much as possible. In the end, with Ottoman army failed to conquer Astrakhan the Ottomans and the Khanate collaborated to eliminate the Muscovite presence or at least influence in the North Caucasus. On the same vein, the Crimean Khanate conducted foreign diplomacy on their own, with little Ottoman influence. The Crimeans frequently allied with Poland-Lithuania to oppose Muscovite Russia, they corresponded with kings and queens of Sweden, and conducted diplomacy with the Habsburgs during peacetime. However, as time goes Crimean foreign affairs become less and less independent and by the 18th century the Khanate could not act without Ottoman influence

In the end, Ottoman-Crimean relations was a reciprocally profitable relationship. For the Ottomans, the Khanate provided them with a strong proxy power and military ally in the north without committing their own resources, while simultaneously providing the Crimean Tatars with the protection of a powerful empire which has one of the strongest militaries in Europe and Asia. Yet, it was at the same time a reciprocally restrictive arrangement. While the Ottomans prevented the Crimean Khanate from gaining too much strength under the fear of its ultimate independence, the Crimean Khans did their best to limit the Ottoman influence, manipulation, and control in the north to protect their own independence or broad autonomy, depending on the time period.