if you have a visum from Switzerland, it is not important if you give your fingerprints. The country which gives a visum is responsible for the asylum application for the whole time of the validity of the visum and 6 months after that. If you go to France and ask for asylum, French authorities will probably find the information about your Swiss visum in the European computer system and make a decision to send you back. The chance to apply against it is not good if you don't have a spouse or children in France or are dependant on some relatives there because of your bad health.
This is a regular situation, exclusions are always possible but rare. I work in Germany and have no information how regularly France applies a Dublin Rule and sends people to other european countries in such cases.