New Vanity: Sink Won't Drain Completely

This might not be the answer, but for reference here is a problem that can occur if the plumbing is not installed properly.

If you look at Fig. 3, this is what a proper drain looks like. You'll notice the orange line represents the water level in the system, the water levels out in the trap preventing sewer gases from entering the sink drain.

In Fig. 1, you can see what will happen if the drain line has to go up before meeting the main drain line. Again the orange line represents the water level in the system, and as you can see the water level is much higher in the drain.

Fig. 2 shows what could happen if the sink was installed lower than the rise in the drain line. The water would never fully drain from the sink, since it would require extra pressure in the system to push the water up the drain.

I wouldn't guess this is the issue in this case; since as you can see, if you opened the trap in this case you would have more water drain than what is typically in the trap.

In a perfect world Fig.1 and 2 would never pass inspection, but as we all know not all plumbing is properly installed and/or inspected. So this could possibly be the case in this situation (given the plumbing was installed by an amateur plumber), but it is doubtful. It would be more likely caused by a clog in the line.

I see one issue that doesn't meet the plumbing code. Before you drain down in the vertical there should be a vent. An air admittance valve will take care of that and probably your dining issue too