Can you give her a soak in warm Epsom Salts bath and then post a photo of her wet back end before you dry her? Cup your hand between her legs when she is roosting at night and do the same to her sisters so that you can compare one to the other. Soiling her butt feathers suggests that there is some swelling. If you insert a lubricated gloved finger into her vent whilst she is soaking, how does it feel.... ie quite tight like there is pressure from surrounding tissue or reasonably relaxed. She may poop when you remove it but this will give her a little relief if she is starting to get bunged up, as eventually happens with these sort of problems. Can you monitor her droppings (perhaps cage her overnight in the coop) and get some photos of her poop) and watch her closely when she is eating because sometimes they will pretend to eat
I agree with
@Wyorp Rock that this is likely a reproductive issue and there is an internal mass which is putting pressure on her respiratory system. The fact that she has stopped laying eggs at 3 years old and you describe her as hefty, are both indications of that, together with the yellow soiling and worsening respiratory distress when you handle her. It is also the time of year when these reproductive issues come to a head because they have been through a dormant period in their reproductive cycle during moult and winter when their body has a chance to recover a little from the effects of a malfunction, but then their hormones kick in and they start ovulating again and whatever ailment they are suffering (internal laying or salpingitis etc) causes the mass inside them to start increasing in size again. Does she have a nice ,plump comb?.... that tells you that she is ovulating. If no eggs are coming out then they are going somewhere else inside her. You will not feel individual eggs in her belly, just a fullness compared to a healthy laying hen which should have a slightly flaccid abdomen. Usually with internal laying and/or ascites, the fullness will be between the legs, whereas salpingitis causes swelling around and just below the vent because the mass is suspended in the oviduct. Sadly if this is her problem, there is not a lot that can be done unless you have a good vet and lots of money to throw at it... please ask for more info if this is the case but surgery or hormonal implants are the two options. If she has ascites, which can be a symptom of internal laying, draining her can give quite dramatic relief, but comes with it's own risks and is only a temporary solution as the fluid will build up again.
Assessing her poop out put overnight will give you an insight into whether her digestive tract is becoming affected/impacted, if there is an internal mass. The internal vent exam will also help you to gauge that. Seeing a photo of her wet back end may enable us to assess if she is swollen.... feathers can hide a multitude of sins!
The other bird that is gaping may have blocked nares and have difficulty breathing whilst she is eating the sloppy food. It should be easy enough to check for that and unblock by gently bathing with warm, moist Q-tips and/or picking them clean with dental sticks.