Genuine Neon Tetra disease is actually fairly rare. It develops over time and kills eventually. If your fish look healthy but are dying after a week or so, then they are not suffering from Neon Tetra disease. Neon tetra disease involves loss of colour, internal growths which cause bulges and eventually quite extreme back bone deformities so the fish looks like it has been folded up - death follows.
Neon tetras are very sensitive to nitrogen pollution, that is why you often see advice not to add them to tanks that have been running for less then 6 months. Is your tank cycled? How was it cycled? How long has it been running cycled? What else is living in the tank? How are they? Without more information, it is difficult to suggest anything.
Having lost an eye is an inconvenience, certainly, but many fish live to their full life span with such a disability.