How to find jobs which are H1B Cap exempt under Non profit category?

Hello ,

I am currently on L2 but very soon my spouse is going to convert to H1B which means that I will have to go unemployed.However, I have read that if i find a job in a non-profit education related employer then, they can file my h1b anytime of the year since they are cap exempt.

So, my questions are:

1.How should I find jobs/employers that are cap exempt?

2.Although I have read on forums that it is perfectly legal to work for such firms as long as they file for your H1B visa.But is there something I should be careful about (with the new employer)?

3.Say if i secure a job today so how long does it usually take the non-profit employer to file and get me my H1B so that i can join them?

Thanks in advance