tl;dr: The corpses would still decay, but probably much slower.
From the Wikipedia article about decomposition:
Decomposition begins at the moment of death, caused by two factors: autolysis, the breaking down of tissues by the body's own internal chemicals and enzymes, and putrefaction, the breakdown of tissues by bacteria. These processes release gases, such as cadaverine and putrescine, that are the chief source of the unmistakably putrid odor of decaying animal tissue.
Prime decomposers are bacteria or fungi, though larger scavengers also play an important role in decomposition if the body is accessible to insects, mites and other animals.
Clearly autolysis would happen also on that planet. So the question is which, if any, of the necessary bacteria and fungi are likely available (from your assumptions, animals aren't).
Wikipedia goes more on detail on the different stages:
Five general stages are used to describe the process of decomposition in vertebrate animals: fresh, bloat, active and advanced decay, and dry/remains.
For the fresh stage, especially the following paragraph is interesting for your question:
The small amount of oxygen remaining in the body is quickly depleted by cellular metabolism and aerobic microbes naturally present in respiratory and gastrointestinal tracts, creating an ideal environment for the proliferation of anaerobic organisms. These multiply, consuming the body's carbohydrates, lipids, and proteins, to produce a variety of substances including propionic acid, lactic acid, methane, hydrogen sulfide and ammonia. The process of microbial proliferation within a body is referred to as putrefaction and leads to the second stage of decomposition, known as bloat.
With your low-oxygen environment (which probably was once a no-oxygen environment) I'd expect especially the anaerobic bacteria to thrive, therefore I'd expect the body to at least go to the phase of bloat.
The description of the bloat phase in the Wikipedia article should be relevant except for the last paragraph, which refers to insects/maggots.
The active decay phase is described as follows:
Active decay is characterized by the per of greatest mass loss. This loss occurs as a result of both the voracious feeding of maggots and the purging of decomposition fluids into the surrounding environment. The purged fluids accumulate around the body and create a cadaver decomposition island (CDI). Liquefaction of tissues and disintegration become apparent during this time and strong odors persist. The end of active decay is signaled by the migration of maggots away from the body to pupate.
Now the maggots would not be there in your environment, therefore I'd expect the active decay phase to be much less active. However note that the purging of decomposition fluids should happen also in your environment.
So in summary, I'd say you'd certainly go into the bloat phase, and you'd probably get a decomposition phase, but not as fast as on earth.
For your cave scenario, also the following paragraph seems relevant:
A body buried in a sufficiently dry environment may be well preserved for decades. This was observed in the case for murdered civil rights activist Medgar Evers, who was found to be almost perfectly preserved over 30 years after his death, permitting an accurate autopsy when the case of his murder was re-opened in the 1990s.
I guess the burying part is only for preventing access by animals, which means the same should be true for the dry cave scenario.
This page also contains relevant information:
It is also important to note that the internal organs of the deceased will begin to decay in a particular order; beginning with the intestines, which as well as holding bacteria also hold various levels of acidic fluid which – when unable to circulate – begin to eat through their surrounding tissues. As the intestinal organs decay so too do the liver, kidneys, lungs and brain. The contents of the stomach may also slow down the rate of decay if there is undigested food in and around that area.
Clearly those intestine bacteria would be present also in humans on your planet, as without them the humans couldn't live to begin with.
Also on this page there's a hint that lack of oxygen slows down decomposition:
If a body is left submerged in water the rate of decomposition will typically be much slower due to the low temperatures and levels of oxygen, unless the corpse is able to float to the surface where insect colonization can occur.