My impression was that members of the actual Nazi party were able to use their influence to get out of the most dangerous assignments. Is this accurate? How many of the front line infantry and officers were actually members of the Party?
There is no general answer.
- Party members who would strut around in safe areas in fancy uniforms were called "Goldfasan" (translated "Golden pheasant"). They were common enough to draw notice and scorn from ordinary Germans.
- Several million men were members of the NSDAP. Not all were excused from frontline service.
All other things being equal, a senior party member would have a better chance to avoid the draft (early in the war) or at least the most dangerous assignments. Civilian workers who were important for the defense industry, transportation, or administration were called "unabkömmlich" or "UK" (translated "indispensable"). Many in the government administration were party members.
Some certainly did, including Waffen-SS personnel, who were often party members, unless they'd been conscripted into the SS. The party had 8.5 million members by 1945, and they could not all have dodged service. It would also have been grossly hypocritical for them to do so, given that Nazi ideology stressed that the individual's highest duty was service to the racial state.
That didn't keep thousands of senior party members, often called "Golden Pheasants" after their smart brown uniforms with golden badges, from dodging service, which created the image. One of their main ways of doing that was to be classified "unabkömmlich" ("UK," "indispensable"), and many Nazi Party officials managed to gain that classification early in the war, although keeping it was harder.
Source: Germany and the Second World War, volume IX/I, page 11. This volume deals with German society during the war.
Many Nazis did fight on the front lines, if for no other reason than that their party status depended on that fact. For instance, the fiercest graduates of the Hitler Jugend were put into the Waffen SS divisions, where they were given the most critical assignments, but also the highest priority for equipment and supply.
Other, more senior members of the Nazi party were "too critical," either in actual fact, or under pretense, to the civilian leadership or management to be used on the front.