I do not understand the lucky part, every foreigners needs to be registered at the address within 24 hours, a law since 1975.
You accept to pay and rent the place, even with knowledge that you will have problems to stay long term in Thailand as this registration is needed for application, OR you say sorry, I go elsewhere where the landlord does register me and avoid problems at Immigration.
It is easy to know. Open your passport, look at the last stamp.
IF that is a small square stamp with Admitted Untill, you can stay until that date.
BUT if you visited Immigration Office and got a big stamp with the text Extension of Stay permitted up to xx/xx/xx, than the xx/xx/xx date must be the last day of working. If that day is in the past your are in overstay since that day until you make your way into the Immigration Office to cancel yours stay based on employment.
IF your daughter is in Thailand during the first 90 days she got on entry using this Non Immigrant O visa or by conversion at immigration, the visa is stamped used and can not be transferred into a new passport.
Just a little advice: DO not give YOUR passport and YOUR work permit to another colleague and hope all get cancelled. YOU signed for these documents and stamps and YOU will be held responsible if things go wrong.
As the 90 day report is also asking for your current address, you can do it everywhere in Thailand AS LONG AS your address is registered at the same Immigration Office.
There is NO rule that forbids you to travel or change address during one year, you are not obligated to make the 90 day report at the same office where you got your Extension of Stay
IF you use the on line method, you can fill in your home address even when temporarily residing elsewhere in Thailand
Show us the proof that you can NOT get a 1 year Extension of Stay after you got a 60 day Extension of Stay based on visiting Thai family.
I understand your story as the Extension of Stay "60 day friends and family Extension" does not exist and therefore can not be issued.
It is a common practice when the foreigner did not have time enough to deposit his money into his account, to gain extra time, first the 60 days based on visiting Thai family than the 1 year.