What is the organic material in Saturn's rings? The phys.org articleSurprising chemical complexity of Saturn's rings changing planet's upper atmosphere about the chemical complexity of Saturn's rings, quotes a member of the INMS (Thomas Cravens) as saying,


  What the paper is describing is the environment in the gap between the inner ring and upper atmosphere, and some of the things found were expected, such as water," Cravens said. "What was a surprise was the mass spectrometer saw methane—no one expected that. Also, it saw some carbon dioxide, which was unexpected. The rings were thought to be entirely water. But the innermost rings are fairly contaminated, as it turns out, with organic material caught up in ice. (italics and boldface are mine).


What is the nature of this organic material?

Does he simply mean a few carbon-based molecules like methane, propane, and butane, or maybe only methane,  or is it a much more interesting cocktail of chemicals than that?

It also links to Chemical interactions between Saturn’s atmosphere and its rings published in Science, but the article is paywalled.
It is indeed a much more interesting cocktail of chemicals.
Before plunging into Saturn, Cassini collected some invaluable data about composition of its atmosphere and how the complex ring system works.
Cassini experienced some &quot;heavy ring rains&quot; in Saturn's innermost D ring, which turns out to be dust grains. The probe tried to use the Ion and Neutral Mass Spectrometer (INMS) to analyse the chemical content of this rain even though it was designed to analyse gases but the rain hit Cassini at such tremendous velocity that INMS instantly vaporized. It was able to collect some data about its composition which include molecular hydrogen, water, butane and propane and trace amount of  methane, ammonia, carbon monoxide, nitrogen and carbon dioxide. A second study was done when Cassini used Cosmic Dust Analyzer(CDA) and inspected 2,700 grains of dust and found that, while most of the material (around 95 percent) was water ice, there was also trace amount of silicates.
The material flowed from the ring downpour was huge. Cassini measured 22,000 pounds (10000 kg) of material per second falling from rings. Most of the chemical slurry is being purged from the D-ring and in turn the C-ring is somehow replenishing the D ring (which is playing a role in the composition of the planet's ionosphere and atmosphere).
Cassini also predicted the composition of E-rings by inspecting the ice grains ejecting from Enceladus. New kinds of organic compounds have been detected in the plumes bursting from Saturn's moon Enceladus. Powerful hydrothermal vents eject material from Enceladus' core, which mixes with water from the moon's massive subsurface ocean before it is released into space as water vapor and ice grains. The newly discovered molecules, condensed onto the ice grains, were determined to be nitrogen and oxygen-bearing compounds. The organic material found were categorized in three types: (i) Amine (ii) Carbonyl  (iii) Aromatics
Additional information of E-ring came from the compositional profile of ice particles from Saturn’s moon Rhea, from a series of spectra recorded on Cassini’s Rhea flyby (R4) in 2013. No striking change in composition is observed but the ice grains had a varying size distribution. Sodium salts and organic compounds were more frequent in relatively large ice grains, whereas pure water ice particles become more abundant in smaller E ring grains.
Source:

https://www.nasa.gov/feature/jpl/nasas-cassini-reveals-new-sculpting-in-saturn-rings
https://www.smithsonianmag.com/smart-news/saturns-rings-rain-organic-compounds-its-atmosphere-180970475/
https://www.sciencealert.com/saturn-ring-rain-heavy-chemical-composition-organic-compounds-cassini
https://www.nasa.gov/feature/jpl/new-organic-compounds-found-in-enceladus-ice-grains
https://www.space.com/saturn-moon-enceladus-organic-compounds-in-plumes.html
Khawaja, Nozair. (2017). Organic compounds in Saturn's E-ring and its compositional profile in the vicinity of Rhea. (link)