Chris - you don't need the mechanical slides, that rail in the center is what the original drawers slid on. If I'm right, looking at that slide from the front, it looks like a "T". Go to the hardware store and spend about $2 for a drawer gide that mounts on the back of the drawer. It's like the mono rail system. They do come in different sises so measure the width of the top of the T the thickness of the top of the T for the size you will need.
I'm building a dresser for one of my granddaughters and I'm using that system because she is 6 years old, but unlike your dresser, I'm adding an interior web (like the one in the picture of Woodnthings' antique repair posted above). The web is a good idea and the lack of it is probably why the original drawers failed. They had to be a bit flimsy not to break that rail, assuming the rail is wood.
As for the opening or drawer size, it's simple preference. You can give it sides that are 1 inch high and deal with things falling out or you can make your sides exactly the same size as the openings and deal with trying to get the drawer into the cavity and swelling and not being able to open the drawer. My preference is to give them a 1/4 inch play.
One more point... those mono rail system guides also keep the drawer from tilting out when pulled past the half way point. With the web system, I cam add a drawer stop. I also plan on adding wooden guides on the sides of the web to keep the drawer from moving side to side (I think that movement is called racking). Good luck
Its' never hot or cold in New Hampshire... its' always seasonal.