I think that on entering Russia, in full parade, with banners, it was one music, and in the camp in the described situation, it was totally another one. In the second case they simply sang songs that French people sang these times. In the first case, it were some official marches, France hymn included.
I was in acting army myself (Chernobyl catastrophe liquidation, a radiometrist) and I was in student camps. Songs were practically the same. Only first-year students sang more often these indecent ones. Youth loves to shock. And elder people (and soldiers mostly are not teens), have no need to shock anybody. Soldiers have enough of it in battle. But... soldiers had little, really very little freedom for themselves. (And nowadays, too, for the main task of sergeants is to make soldiers busy) And if they have an hour for their heart, they wanted to be normal men, if only to be them more intensively (too little time for everything - remember?). So, when they were at rest, their chose more "strong" songs. Intensive for their feelings. But mostly these were civil songs, for they want to forget the war at least for the time of rest. As I remember my experience, often they were about love. Even more popular were joke songs.
And later, after war, when some of soldiers will return home, they don't like marches. They don't like to recall the war. You can't imagine, what pain it is. The only way to recall the war and not to got crazy, is a joke. Or joke song, of course. ... And the songs about the return home. If written by some soldier, or folk ones, they mostly are sad. On the contrary, these, written by half-official poets, are full of energy, but are never sung out of marching. (they are often chosen as marchs). Marches were used for setting the rythm of ehm.. march. They are useful. And they and only they can be considered as specific soldier songs. Because to sing them is the part of their work.
In Russia there are really beautiful Cossack's songs. Cossack's are free soldiers. Their songs are often only somewhat warlike, and are widely known and sung. But in the acting army, never had I head somebody to sing any war song or even a Cossack one. I haven't even seen a person to hear such song by radio. On the contrary, as for students, about 20% of our songs were about war, or soldiers, or some of Cossack repertoir. War songs are for civillians.
And when a soldier murmured quietly something cleaning his gun, it was the same song that he murmured making shoes or drying hay, sometime in the past.
All of said is about real soldiers, that fought. As for rear and staff officers, intendants and such people, THEY like to look maximally warlike and to speak about battles, courage and things, and to sing brave songs. That adjutant that you had mentioned, surely, loved war songs and his friend officers made or even asked soldiers to sing them for him. There are songs, that could be chosen by some troop as their beloved one. Don't forget, that often it was the officer or sergeant who decided instead of soldiers what is beloved for them. Don't mix imaginary romantics with the real life.
As for French soldiers in Napoleonic wars, their troops for the first time in history mixed people from different French regions. And they exchanged songs of their home and created a new whole-France folk culture. But they didn't brought it back from the Russian winter. Most of these few, who remained alife, remained in Russia afterward and were mostly individually separated. They never created such communities as Jews. Only few songs could remain in human memory after that. I would look in the old Russian books of French songs from the start of 19 cent. But surely, they are not on the net. Maybe, it is possible to find on the net the names of such books. And later somebody can try to look for them in some special part of a good library.