I don't think things will work quite the way you are thinking. For example, while a MTB with 26" wheels might sound like it should sit lower because of the wheels, the bottom of the frame is not as low as the change in wheel radius would suggest. This is because there is a smaller "bottom bracket drop" dimension on the mountain bike. This means that the bottom bracket does not sit as far below the height of the wheel hubs as it might on a road bike.

Also, on a MTB, one typically uses a frame that is numerically much smaller than a road bike. This is because the seat tube is much shorter than the top tube on a mtb. And you'll want much more standover on a MTB to accommodate putting your foot down on uneven ground.

As for a road bike, there the frame may be closer in concept to your New Yorker. Not really sure about that, but they seem more similar (except, of course, the handlebars, assuming you don't have road bike handlebars on your New Yorker). And the handlebars may be a significant issue: imagine replacing the bars you have on the New Yorker with road bike handlebars mounted on the same stem and you get the idea of where you'll be reaching.

For an idea of road and MTB frame sizes that may fit best, you could use an online calculator like this one as a starting point: Bicycle Frame Size Calculator. You will see a much smaller MTB frame size is called for than for road. Interestingly, you will find that the road frame size suggestion is smaller than your current bike but your current bike works for you, perhaps partly because you don't have road bike handlebars on it (at least, I'm assuming you don't).

The reality is that you will not know for sure if something works for you until you ride it.