unscrew the divertor end cap and remove screen
An aerator is on the end of your kitchen or bathroom faucet spout that unscrews and holds the screen and/or water saver.
The shut off valve under the sink, the faucet itself, or most likely, the screen in the end of the spout is plugged. If neither hot or cold is coming out, it is kind of unlikely both shut off valves and both sides of the faucet would go bad at the same time. The screen being clogged would stop any water from coming out. The screen just unscrews out of the spout.
There may be debris in the pipe from adding the new pipes that have plugged the screen in the faucet. The end of the spout has the screen and unscrews. The valves themselves can also be plugged.
The screen in the end of the spout is probably clogged.
At the very tip of the spout.
Probably sediment in the valve or the screen in the end of the spout. A piece moving around and sometimes blocking the flow and then falling back when you turn it off. Try taking the screen off the spout and see if that helps. The end unscrews and wil have a screen and aerator in it.
Possibly unscrew your aerator at the end of the spout and see if the screen is clogged. Blow it out and rescrew the aerator on. If that isn't the problem, sometimes things get caught in the spout. Unscrew the spout and turn the water on. If there is water coming from there and not out the spout when it is hooked up, the spout is plugged. If no water coming out when the spout is off, rehook spout and change cartridges or stems in the faucet AFTER shutting the water off. Depends on the brand of faucet, check the instructions for the brand and model. Also your kitchen dishwasher and sinks are run on seperate valves. The dishwasher valve is usually under the sinks (causing the water to go to the dishwasher first) the valve that was installed for the sink (before the dishwasher) may be bad too.
Take the screen out in the end of the spout. Lots of times, minerals or rust in the line will break loose when you shut the water off and work on a line and plug the screen. The new faucet may have a flow restrictor in it that the old one didn't have.
Normally at the end of the faucet spout
Spout removal tool made by Ridgid or General
You say "a" pipe so it is the spout? Normally you would cut it to the right length or replace it with a new one. There is no flange or filler piece for this. Can you get to the back side and set the faucet deeper. Usually there is a 2x4 holding the faucet in place. Possibly on edge and notched out for the faucet, cut the notches deeper.
I just installed an American Standard kitchen faucet and it continues to run for a second or two after turning the handle. I called AS and they said it is because of the high arch and the location of the stop valve lower in the fixture. I think it may have more to do with this being a lower end kitchen faucet. During a bathroom remodel we installed a high arch Pegasus bathroom faucet and, while it is admittedly not at high as the kitchen faucet, it stop immediately and doesn't continue to run/drop at all. When I removed the Pegasus faucet filter at the spout, it behaves just like the kitchen faucet, so there's some sort of stop valve element at the spout of the Pegasus that isn't part of the AS faucet.