in your question there is already a discrepancy..if you got a 3 months non B visa ..one company has provided all the papers for you to get it and you must get a work permit with this company otherwise your NON B visa is VOID ..CLEAR ENOUGH...if another company is willing to process another non B then you have to restart from zero unless you gett a work permit first with the company that provided you the first NON B and a letter from that company that they are fine with your resignation and allow you to get another job in this case you can change the employer on your work permit.

You can stay on your non O

NO by law you must go back within a week but in reality the immigration do not enforce this law ..they let your visa expired and do not bother you but you can not renew it ...you must have kids if you need another visa or older than 50 and get a retirement visa. I know because one time I have asked.