when my cats are falling asleep, the 3rd eyelid comes out and sometimes stays out a while if I wake them up before they want to be awakened. but with the already illness in your home... If you can't afford an office visit, ask them if you can pay in instalments? Or call a local rescue org and tell them you are poor and tell them your cat is a rescued stray (just say that) and needs vet care- they can get you an office visit for 10 bucks or zero and meds at cut rate if you say the right things.