Had the exact same thing on my Audi A4 a few years ago its fuel pump was gone and when it got replaced the new one blew the whole fuel system. (Or so the Audi garage told me)
if its 81 or 66 kw engine it doesnt have fuel pump in the tank. It only uses suction of the injection pump. if it shakes on idle, i would say the dual mass cluch is shot.
2023 2.0 tdi octavia after 110 km/h and up vibrates the stering wheel and whole front of the car. 2 diferent cars had and they have the same problem it started from 40-50 000 km. Skoda changed the brakes komplet, the antrieb and the transmission ona after another. Problem still here. No solution.