Do I need a vent for extending a drain to a new bathroom addition?
You must have a vent above the vertical run for these drains. As drawn, water from your new work cannot draw air behind it from your existing work.

This is what vents do: draw air in behind draining water.

You could functionally combine all the new work drains as drawn into a single vertical vent, but remember that it needs to be able to draw air from above their combined drain line.

Depending on your code you may or may not be able to combine these into a single vent, but they will need their own vent.