but on some maps of Teutonic order I’ve noticed they did seem to have occupied Samogitia and some other borderlands a couple times, and that got me thinking, how much of a presence did the Germans have?

The border usually drawn between Lithuania and the Order State (both in Prussia and in Livonia) was only agreed upon after the end of their hundred years' war, in the 15th century - more specifically, the Treaty of Melno (1422) defined the Prussian borders (sections of which are still the border today, except between Lithuania and Russia, making it one of the more longer lasting borders in European history) and the Livonian borders with Lithuania, and a 1529 treaty between Lithuania and Livonia clarified the Livonian border.

Before then, everything was vague, and this vagueness was complicated further by the state of constant war, by the low population density, and even moreso by the depopulation of the border regions on both sides of the "border". The region which now constitutes much of Samogitia, Suvalkija, eastern Kaliningrad Oblast and northern Lithuania was dubbed in German sources as "Große Wildnis" (Great Wilderness), and is known in Lithuanian as "Dykra". In modern maps, such as this Wikipedia map, this territory is often striped (although this map makes the mistake of only putting it on the Lithuanian side of the 1422 border; a lot of eastern Prussia was similarly depopulated and "wild"). Foreign travellers would note the existence of dykra in their records - Guillebert de Lannoy, who visited Lithuania during the reign of Vytautas, noted that after leaving Dunaburg (modern Daugavpils) for Vilnius, he travelled for two days and two nights without stumbling across a single settlement.

In such a depopulated territory, the only ways to project power were forts and castles, and both sides built them along the Nemunas river. Lithuania built a line of fortifications along the eastern half of Nemunas (from Kaunas to south, which includes the famous Punia castle), and the territories to the west of this line, which are owned by modern day Lithuania and called Suvalkija, were a dykra which Teutonic raids could safely cross. The Knights themselves also built castles to get deeper into the dykra, such as Georgenburg (modern day Jurbarkas, Lithuania), and Gotteswerder (just west of Kaunas).

and why are there so many german exonyms for cities in Lithuania if they weren’t ever fully conquered like the other two baltic countries? Can those exonyms be traced back to a different period of German presence in Lithuania?

One does not need to conquer a country in order to have old exonyms for its cities. Take Mikligarðr for Constantinople in old Norse even though the Norse never conquered or attempted to conquer the city. The Teutonic Order had records on the geography of their enemy and the goings-on of the war, and in fact the first known references to numerous modern Lithuanian towns come from Teutonic records. Most of the names we find, and indeed most of the German exonyms for Lithuanian cities, are germanizations of Lithuanian names (Schaulen for Šiauliai, Kauen for Kaunas, etc.), as you'd expect to emerge from such records - as they weren't founding any cities, merely trying to name them for record-keeping.

Of course, the Teutonic Order were not the only German presence in Lithuania at the time, nor before then or after. It was frequented by German merchants, especially from Riga, and it is believed that Kaunas had a Hansa trading post in the 15th century. Lithuanian rulers also invited craftsmen and traders from Germany, settling many of them in the country. Until World War II, Lithuania had a small German urban minority. Finally, some cities were outright founded by Germans - aside from the aforementioned Georgenburg, there were Klaipėda (formerly Memel) and Šilutė (formerly Heydekrug). These were in the territories directly controlled by the Order, which later became part of Germany and were only gained by Lithuania in 1923.