Me too. Others on the board who've taught in multiple provinces probably know better, but I recently got a Z visa, FEC, and RP in Guangdong, which all go together. When you changed schools from Harbin to Zhuhai, I would expect that your new school would have to get you a new FEC from Guangdong, as well as a new RP from Guangdong within a certain time frame, but I'm not sure. Back in 2005, I know that in Guangdong you could get an F visa foreign expert cert. that allowed legal working (not the same exactly as the current FEC booklet), but that was a loooong time ago. Might be a way to get around the residence permit and Z visa.
You might want to give some of your basic info like nationality. Everything I have been told the past two years is that Guangdong will not allow foreign teachers to go to HK to get a Z visa unless the FT shows residency there. The Z visa needs a couple of letters of invitation from the school that give the home address of the FT, and the Z visa can only be gotten from the Chinese embassy/consulate in the country listed on those papers. If the school will put HK there (there are quotas for HK) or maybe Thailand or something, then you could avoid going all the way back to UK/USA/Canadia, wherever you're from. My school wouldn't do it.