Good grief, what a tiny cage! That poor thing. Sun conures are pretty close in size to nandays, I think, so imagining my baby in that cage is just heartbreaking. If I lived anywhere near Dallas, I would take him for sure, but I'm nowhere near Texas. I really hope you get him!
As for health problems, I imagine he's suffering a lot from a lack of essential vitamins and minerals. He probably has a vitamin A deficiency. Vitamin A, after doing research, helps with mucous production. The mucous is made to keep microorganisms from infecting birds, so with no mucous, he's susceptible to infection. Apparently a bird's respiratory system is most likely to become infected, but so many microorganisms can take root in a bird so a lot of things could be happening with him at the same time. You'll have to watch for symptoms like sneezing, wheezing, nasal discharge, diarrhea, feather color deterioration, weight loss, lack of appetite and other symptoms, I'm sure.
I can do other vitamin and mineral research for you if you'd like!
I wouldn't load him with vitamins and minerals quickly if I were you. It sounds like a good idea, but giving the normal dose of vitamins and minerals to any creature starved of them can have disastrous results. Slowly acclimate him to a much better diet.