What is the push button 4wd buttons plugged into?

That black box is part of the switch assembly itself best i can figure. As for 'now', you now need a new switch assembly! All you have to do is simply unplug this switch! Where the wires go is not relevant to what you are doing.

It's the transfer case shift control switch. It plugs into the transfer case shift control module. It's mounted under the right kick panel.