I can't help you with NYC, but I think your budget is going to limit your choices there. A NY agent will be able to help you out there.
I can help you with NJ, though, and I could tell you about some townhomes, that are condos, not coops, that would stay in your price range. A town called W. Orange has some lovely complexes, and some of the spacious units are now priced under or around the 400,000 price point. The complexes offer pools, tennis and clubhouses with exercise and meeting rooms. Some are walking distance to the NY bus.
My question to you, is - what do you consider to be a "short" commute? From my area, the bus to Port Authority will take about 50 min.................the train 35-40 min, but parking may be an issue if you don't live in the "train" town. Another option is to drive to Harrison, and take the Path train (20 min drive).
Are you looking for a city-type environment, or a more suburban feel?
Hoboken and Jersey City will offer a more urbane feel . I don't cover that area, but I am sure agents will come along who do.
If interested, please reach out to me, and I can send you some listings to give you an overview of my area.
Best wishes in your home search - you sound like very qualified buyers, and I am sure you will find just what you are looking for!
Prudential NJ Properties