You quote having a total of 4400 hours of which 4200 are on 737's presumably as copilot. This suggests you were fortunate enough to walk into a 737 job with only 200 flying hours under your belt.
I think the trouble you will strike is that both in Australia and NZ it is doubtful if any general aviation operator will give you a job on a light single or twin since you lack experience apart from twiddling the autopilot from the RH seat of a 737.
This is not to criticise your experience as an airline pilot but apart from extremely expensive cadet schemes where you start from scratch with that airline as a trainee, there are already a surfeit of experienced general aviation pilots in Australia and NZ, waiting in the endless queue for jobs as copilots.
Your basic CPL 200 hours (forget the 4000 hours copilot on the 737 as in these countries it is worth very little in terms of real decision making experience), puts you well behind the others who may well have over 1500 hours command time on other types. Stay where you are and build command hours on the 737. Your future is then assured in countries that need 737 captains. Australia and NZ are both relatively minor players in aviation generally compared with the huge aviation industry in SE Asia. Both OZ and NZ are pleasant places to live although some see living in Europe more attractive in terms of lovely countryside. And probably more flying jobs too for someone of your experience.