I would say fuel. But its hard to tell from words on a screen.

Check the fuel pump pressure and flow.
Also check for kinked or crushed fuel lines all the way back to the tank, even with a new fuel pump this could be the problem.
And check the fuel tank is breathing too. Maybe leave the cap off when driving to see what happens. More of a problem on bikes but you never know.

The carb may need attention.
See if the butterflies are moving around, the shafts may have worn at their mountings making them free to jiggle around a litle bit.
Check the cable play, throttle and choke, could also be loose.
Does pulling the choke out fix the surging when driving?

It could also be an air leak. Check the carb is tight on the manifold and also the carb is screwed together tightly. Grab it and give it a wiggle.
Also spray the carb with water when the engine is running and see if the revs change. There could be an air leak in one of the gaskets.

When the car starts surging, switch it off, roll to a stop and check the plugs to see if they are fouling. Black plugs means the fuel/air mix isn't burning properly. It wouldn't hurt to check the fuel level in the float chamber here either.

Checking for looseness in the cables would be my first guess, it sounds like they could be flopping around under half throttle.
Good luck.