Moving to HCMC in the near future

Hi all, myself and my girlfriend have recently traveled through SE Asia and we have just started looking for work teaching English in HCMC. We are currently in Phnom Penh, but we are thinking of going back to HCMC because I would imagine it'll be easier to find work if we are in the city. We both have degrees and CELTA's and have been advised getting a job shouldn't be too difficult.

Quick visa related question. We have been advised that if we come to Vietnam on 3 month tourist visa's we will then have to do a visa run if an employer applies for a work permit on our behalf, is that correct? We were advised that if we get a business visa we don't have to do a visa run, but as far as I can see we need a letter of approval to get a business visa.

I know this is a very broad question, but what district would you recommend living in? We want to be centrally located(ish) and will probably be looking at spending approx $500 on a 1 bed flat

Thanks in advance.