Not sure what is wrong with my girl

Hi All - just signed in as a new member because I haven't been able to find questions with the similar symptoms of one of my girls. She was born this past March and brought her home April 4th with 5 other chicks. She is a rhodebar. We are down to four total because of a friendly neighborhood owl that was visiting one day. The other three are a rhodebar, black star and red star. They seem fine. This is what I have observed. This one (Dale) has always been the smallest of the four and not filling out as nicely. Once they started laying someone layed 3 soft shell eggs. I'm thinking it was her because since she's been separated all eggs in coop are good. Four days ago she was very lethargic and when they came out of their coop she layed down under a bush with her one wing stretched out. No interest in eating at grass or even walking. She seemed to keep her feathers puffed out like a big round ball. Palpated her vent and did not seem egg-bound. Used K1 and a gloved finger and inserted as well but there was nothing there. Warmed her vent area with warm wet towel to try to relax her as well as bedding her down on a warmed towel. Next day still very lethargic so set up a dog crate in our screened-in porch. Thought might be peritonitis because of soft shell eggs that had been layed, so started her on 100 mg doxycycline given in 1 teaspoon of water in a syringe. Gave her 1/2 tsp 2 x day. By next day she seemed perkier. This is where I am confused. She seems to sound phlegmy when I give her the meds or she tries to eat something. Sounds like a sneeze/cough saliva bubble. She doesn't seem to have much interest in eating dry pellets, but appetite is increased when pellets are watered down into mush and loves eating pieces of grapes. This is day 3 of doxy. so don't know if or how long I should continue it if this is a cold-type problem although there is no runny eyes, snotty nose or any other signs of a respiratory problem. Could she have an issue eating dry pellets which is why she has always been smaller than the other four and what would the saliva-sounding cough/sneeze be only when she eats or drinks?? Appreciate any help I could get - not sure if I should put her back with the other girls if all signs of her look good now except for this little funny cough sound.