Yes, it's definitely true that experiencing stress makes cheilitis worse. Stress makes almost every physical and mental malady even worse. It's so destructive.
I also had times when the damaged upper layers of my lips turned mushy when eating and were left discolored.
I think that the mushiness is from saliva and chemicals in the food we are eating getting onto the lips (which cannot be helped, when one is eating) and chemically breaking down the weakened skin so much that it turns to mush.
I do think that it would help your lips if you washed them with a very gentle soap, such as the one I am using (the EO organic hand soap, peppermint/tea tree variety) every time after you eat, and also every time after you brush your teeth.
The soap wash gets all traces of the other substances off your lips so they don't sit there invisibly for hours after you eat and weaken your damaged lip skin further.
If you don't want to wash your lips with soap, then at least rinse them with warm water every time after you eat. This will entail going to a sink, because you really want to do a good rinse (a napkin dipped in a glass of water isn't enough).
And after your periodic washes or rinses of your lips, please try a tiny dot of the glycerin - I got a 6 ounce bottle for $2.50 at a drugstore and I've still only used about 1/2 of an ounce, so it's very economical and it will not hurt your lips in any way. Either it will help, or be benign. Even on the bottle, it says it's for treating chapped lips.
A tiny dot of glycerin is all you need, spread it around, give it a minute to dry, and then no one will know it's there. It kind of glues down the dry skin, so the lips appear more integrated and less flaky.
It also makes your lips a little bit sticky, perhaps a better word would be "tacky" (in the glue sense, not the "bad taste" sense! ha) so for me, that tackiness of the lip surface discourages me from rubbing my lips together, which I did when I used vaseline or oil on my lips (they made my lips slick and I would want to rub them together).
I'm not saying that washing with a mild soap and using glycerin as a moisturizer is a cure, but it's really good at management of the lipskin and keeping it clean, healthy and hydrated (without being over-hydrated or over-oiled).