the end and the beginning of a responsibility to take an asylum procedure of a person is always a very difficult question with the refugees going from one country to another.
In Article 13 Part 2 of the EU-Dublin-Regulation there is a rule, that if you stay in any Dublin country (after leaving another country which was responsible before) for more than 5 month without a break, this country will become responsible for processing your asylum application. But it must be clear that you really were in this country during this period, which may be difficult to prove if you stayed illegally.
To make a more exact assessment you must speak to a good councelling center or to a lawyer with the expertise in Dublin cases showing your papers, earlier decisions etc. But im principle nobody will tell you, if Austria takes you back again after the request of Switzerland or not.
Even if Switzerland takes your asylum application, it probably will be proceded as a so called second application (checking only if you have new reasons after rejection in Austria. If you can't prove that your situation in your country of origin changed to worse after this first rejection, you can get a rejection as inadmissible. Naturally I don't know exactly how the asylum law in Switzeland works, it is not EU and there can be some differences.