Can you install a ceiling light in addition to the over-the-mirror light?

Recessed lights in the ceiling is what I would do.

Are the shadows you object to beneath the medicine cabinet and the cabinet to the left of the sink?

No, the problem is more that my face is somewhat in shadow when looking in the mirror

Can the light fixture be moved up on the wall so that you can flip it and direct the lighting down?

I have done a box before. In my case, the light fixture ended up on the slanting part of the ceiling and the fixture could not accommodate the slant. So I had my carpenter build a box to match the slant and make a level place for the light to attach. So you can have a box made for behind the light fixture. Others have posted here, asking for help finding a light that protrudes enough, and I don’t think any have been found.

Can lights have their own issues above a vanity. They usually make shadows on the face of the person shaving or doing make-up. Light behind you may help. These ceiling lights are very easy to install and they send out the light 360 degrees around, plus, their color can be adjusted. They might help in your situation. One over the shower is nice, too. I found them when doing my bathroom and loved that they need no electrical box behind them and take up only the depth of the ceiling drywall. My builder had used them before and also loved them. You don’t have to worry about ceiling beams as long as the electrician can get the wire to them.

https://www.build.com/dals-lighting-5006-cc/s1763172?uid=4163442

Maybe different light bulb covers???

which shadow do you mean? If it's the medicine cabinet casting a shadow, then finding a way to point the fixture down might make it worse (because the bulbs will be closer to the thing causing the shadow.)

tough love: the easiest answer is to ditch the medicine cabinet and get a flat mirror. then you can flip the fixture any way you like and it should be fine. besides, I think the differently sized cabinets look weird next to each other. (sorry!)

with some creative storage, you can find places for all your stuff. you have a cabinet to your left. put your "getting ready" stuff in there, and just keep the door open during that time.

if storage is really tight, remember that not everything needs to be kept in the bathroom. spare soap, towels, and first aid items could go in a different closet. use the bathroom storage for "frequent use" items only.

I would remove the medicine cabinet and replace with just a mirror. IMO, I don't like a medicine cabinet in front of a sink -- stuff can drop out and fall into the sink. It looks like you have a nice cabinet adjacent to the sink for holding your stuff.

I know this sounds crazy, but I would get a lower profile med cabinet. It seems bulky for the space a bit anyway... another thought is those clear glass globes are a nightmare as they get foggy one, and to they let the light spread rather than focus. I would look for frosted glass down facing. https://www.overstock.com/Lighting-Ceiling-Fans/Kichler-Lighting-Pocelona-Collection-2-light-Chrome-Bath-Vanity-Light/13325181/product.html This fixture projects out 6.5 inches.

https://www.overstock.com/Lighting-Ceiling-Fans/Capital-Lighting-Baxter-Collection-2-light-Chrome-Bath-Vanity-Light/11684248/product.html?recset=86809dc8-bbfa-4993-8b17-a6145ab1b0df&refccid=RSN74YY23SHZ3TKGRDYL663ARI&recalg=966,962&recidx=19

this one looks good projection wise too.