Unfortunately if your chickens have a chronic respiratory disease such as mycoplasma gallisepticum (MG,) it will be with them and your whole flock for life. MG can prevent them from laying well, reproducing well, and they may have shortened lives. It can be passed through their hatching eggs. That is tough to start out with sick chickens. All of your future birds will get exposed eventually. Tylan 50 injectable and tylosin powder are common treatments for the symptoms, and there are a few other antibiotics as well, but they will still be carriers and may get sick again in the future when they become stressed. MG is common in some backyard flocks and in wild birds, but not a good way to start out with your first flock.
There are 8 or so common respiratory diseases, and the best way to get confirmation is through testing or with a necropsy by the state vet if you should lose one.
@casportpony lives in CA, and she knows where to get antibiotics online. She might be of help. Your chicken might get better without antibiotics. You could use saline or eye wash to flush the eyes, and apply plain Neosporin ointment, Vetericyn eye gel, or Terramycin eye ointment if you can get that online. CA has more strict rules about antibiotics than most states. Sorry that you are dealing with this. This is a good article about MG to read:
https://extension.umd.edu/sites/extension.umd.edu/files/2022-04/FS-1008 Recognizing and Preventing Mycoplasma gallisepticum (MG) Infecti....pdf