Ho Chi Minh or Hanoi??

Not a fan of the cities you list. Da Nang was my favorite. Not sure about teaching jobs there though.

Please be ware of some of the school cheating the ESL teacher like you and your boyfriend.

I am familiar with both cities and in my opinion, you should go to Saigon because:
1/ the traffic, street condition and law abiding in Saigon is better than Hanoi.
2/ the people in Saigon are friendlier than Hanoi especially when you are lost and asking for direction.
3/ To me the weather in Saigon is better than Hanoi too.
4/ As a Vegan you definitely should live in Saigon since the veggies somewhat more available/cheaper/less contaminate with toxic. Be sure to do your own reseach on this matter.
5/ It is closer to Thailand and quite cheaper with many good deals offer.

Good luck to you two.

Definitely Ho Chi Minh!

If you like cold weather, Hanoi, if not Saigon.     But in my opinion, DaNang is the best city I have visited and if I move to VN, it would be my first choice, period.

There are less job opportunities in central Vietnam. I sense that you would be better off lining up a job in those cities before moving there. Compared to Saigon where teachers often recommend just showing up and interviewing in person.

Saigon is a big sprawling city, farther out it's surrounded by marshy areas and farms. Pretty far from 'nature' if you mean open space, though you might like the rivers and coast. The south is also very flat, at least an hour drive to hills in the direction of Cambodia. Hanoi much closer to hills and mountains.

Cycling is basic transportation for kids and teenagers. But there are also some serious early morning and weekend recreational Vietnamese riders here, surprising to see Colnago, DeRosa frame-sets costing more than motorbikes. I lived along the river and saw packs of them on Vo Van Kiet, big fast road without cross traffic. Another popular location is in suburban District 7 on Nguyen Van Linh.

You need to have enough money with you.
It is not like Africa extension of visa is possible at all times within the country.
After first 2 times within the country,  the third time is not possible  within thr country. That is to say,  you have go out from the country eg. Cambodia or Laos. All is about time and money. If you are for work,  apply for working visa for at least 3 months before coming to reduce your hassles. Choose Hanoi for grabbing quick job. Ho Chi Minh is conservative. Your skin colour will matter. Less opportunity for dark skin. Nice people but new and novice in globalisation and not used to dark skin Africans as English speakers. Gradually,  a few are making way through and so forth.
Good  luck.