You can adjust the pop-up drain under the sink (behind the drain) where it attaches to the down rod. It usually has a set screw, or a spring clamp that let's you switch holes in the downrod. Some of the cheaper big box brands (such as one that rhymes with "shampton bray") may not adjust very well, esp. if they're lightweight and have a rubber gasket. A last resort is to replace the pop-up assembly with a better/heavier one.

I had that arrangement in two places I've lived, both with the drain in the right rear, but the DW on the left in one and on the right on the other. I think I prefer the drain on the opposite side of the DW (on the right in your case). The dirty plates and glasses get placed into the sink towards the left side to keep the drain area clear, which make it a shorter reach to the dishwasher. But perhaps a more important consideration is which way you find more natural to sweep food waste from a dinner plate into the disposal - if the drain is on the right, I pick up the plate with my left hand, tilt it towards the drain, and sweep the food off it (if necessary) with a fork using my right hand, using rightward motions. I somehow find this more natural than reversing hands, or not reversing hands but sweeping food leftward with my right hand. The kitchen with the DW on the left had an Elkay LWR2522R sink in it, which works great if you need to use a drop-in sink (usually for a laminate countertop) because the recessed drain makes dumping your food waste easy - it always finds its way to the drain and keeps the rest of the sink clear. A left-drain version is available too, but I haven't seen an undermount or built-in sink with this design. There are several undermount sinks with rear-corner drains though. I prefer these to the rear center drain, in part because it also moves the disposer to the back corner of the cabinet underneath, freeing up more usable space to the side, rather than splitting the available space into two uselessly small sections (assuming the sink is relatively centered in the cabinet, often the case but not always). On the other hand, many sinks with the rear-center drain are sort-of D shaped, leaving space for a single-hole faucet (or instant-hot or soap dispenser) on either corner, and allowing the rear wall of the sink (and the drain and disposer) to be closer to the back wall, making the sink a few inches deeper front-to-back. These are available from several manufacturers, including undermounts. Yet another choice: a rectangular sink with a rear-corner drain and no faucet deck or corner cutouts, and a wall-mounted faucet - that gives both maximum sink space and allows for a center-mounted faucet (which I prefer to one in the corner personally). Ah, the minutiae of designing kitchens....

There are a few easy DIY places to check if you're so inclined. Turn the disposal off at the breaker first. -Pull the rubber flange out of the bottom of the sink. You should easily be able to see around most of the interior cavity to make sure nothing is down there. And FYI, this is a good thing to do periodically in the future, both to check the cavity and to rinse off crap that might be stuck to the bottom of the rubber flange. -The hose from your DW into your disposal should have a wire clampy thing over the rubber hose over the inlet part. Pull that off and see if there is any blockage at that point. Wait, have you even run your DW yet? If not, never mind. -Check the P trap; perhaps something is stuck down there? You know, nuts, bolts, rolls of PTFE tape... :) Anyway, it's at least something to check while you wait to get a plumbing appt, which probably won't be until after the long weekend, unfortunately!

Hopefully you have PVC or some similar plastic pipe. The proper thing to do is to pull the vanity, open the wall, raise the pipe, reinstall the vanity. The picture you've drawn begins to look like an S trap. These are not code compliant since they can lead to emptying the trap by siphoning.