Yes, electrons are at all times in random motion proved from the fact that the material has a macroscopic temperature. The kinetic motion we define as thermal energy and measure as a temperature comes from the random motions, fluctuations, vribrations etc. of the constituent particles of the material incl. the electrons.
With random motion of electrons in all directions, plenty of charge passes through any cross-section of the wire all the time. But statistically since it presumably is truly random, equally much passes through from either side, resulting in no net charge flow. No current.
Then you (with a battery or other voltage source) apply an "electric push" from one end of an ideal zero-resistance wire in the form of a potential difference across it. This superimposes a drift on top of the already present random motion. This drift constitutes the full net flow and isn't cancelled out - this drift is macroscopically what we measure as a current.
If you after having established a net drift - a current - suddenly remove the potential difference again, then as per Newton's 1st law the drift will theoretically continue. In the truly ideal scenario, this established current will never end but will flow forever. This is called a superconductor.
No real-life wire is a superconductor*, so there will always be some resistance causing the drift kinetic energy to gradually disperse away from the electrons and into the material (via momentum transfers from bumps, impacts and other interactions with the material that carries the resistance). This typically happens basically at an instant** so that the current stops immediately when the potential difference is removed. But the random motion is still there and hasn't been influenced by all this.
* ... unless you are experimenting with cryo-cooled materials; superconductors can be achieved to some degree at very low temperatures and is quite an interesting topic.
** ... as long as you aren't dealing with time-delaying components such as a capacitor.**