Hello Mireille
Thanks for your post. 19 years is quite a long time so my four years in Britain seem short in comparison. Although I am very happy with the move to Switzerland I still often miss the UK and London. But despite me loving british life and culture I moved indeed last year for work purposes (but also for family as they are not far)
Moving to Geneva without a job is very difficult. I actually even believe that you cannot live in Switzerland if one member of the couple at least do not have first a promise for a contract of employment in Switzerland as this is what gives you the famous B permit (or G permit if you decide to live in France voisine which is the permis frontalier). If one of you has a job offer then the company will deal with swiss authorities to get the permit. You can then move to Switzerland and the partner who has not a job yet will have a temporary 3 months residence permit (you can renew it) until he/she finds a job and also gets the work permit. Bottom line is that you cannot move and live in Switzerland without a job. But as a EU citizen you can come and live in France voisine and start looking for a job from France (if you are considering to move to Geneva of course) which may be easier (and cheaper) in your case.
As you see quite complicated but not impossible. I got transferred by my company from UK to Geneva so it covered the work permit issue. Luckily my partner found a job quite quickly in a few weeks. We found her the job from the UK by doing two or three day returns between London and Geneva for the interviews. So it shows again that it is possible to find something from overseas but I can think that it is still complicated and that we got very lucky. But this is completely possible as I say.
Usually there are plenty of jobs and recruitment agencies in Geneva (of course depending on your fields) so it seems to me that it does not take long to find something. Companies are used to foreigners (they are rarely Swiss companies themselves actually) so I do not feel that they favour locals vs. foreigners. Clearly you being French will be a massive help if you look for a job in Geneva.
I can recommend you a very good book that you can order on Amazon France that really helped us in all our "demarches". The title is "Travailler et vivre in Switerland" from David Letterman. It is well written, very useful and will answer many of your questions regarding the move.
I hope all of this info helps. I am quite happy to share or help more if you have other questions. So please do not hesitate to come back to me if you need it. I hope that you will be able to progress in your project despite all the Swiss bureaucray. Life here is very good (although quieter than in UK) and I recommend it.
Hope to talk to you soon
Eric