You are asking people who are offering temporary tourist accommodation to sign a document attesting to the fact that you live there. I'd run a mile too.

The usual solution to this is to either rent a furnished apartment short-term or move straight into shared (e.g. student) accommodation, depending on your life stage. Then you register, get your work/residency permit and only then do you start hunting for your new home if you still want an apartment of your very own. Lausanne has a >99% occupancy rate, so this last process can take a while; 2-3 months would not be unusual.