How to do about this vanity that won’t sit tight to the wall due to trim molding?

I see a couple of issues. The first is that your three bathrooms are not wired properly, and you should take this opportunity to fix them! You should not have outlets in three different bathrooms all on one circuit. Each bathroom should have its own 20 amp circuit. The reason for this is that hair dryers, curling irons, etc. use a lot of juice.

Your second issue is going to be the ability to "T" off of an existing cable at some point below the existing outlet. Assuming that the cable actually does run down the wall below that outlet, I would imagine it runs in a fairly straight line without a bunch of slack, right? Unfortunately, in order to "T" off of it, you would have to cut that cable and then have enough slack to pull 6 inches or so from each side of the cut into a new box, where you would join those two ends with your new cable. It's highly unlikely that your wire has a foot of slack in it that you could use for this purpose. If it did have a foot of slack, then you would be better off just pulling it through the actual hole for the new outlet and connecting the two ends (from the top and bottom of the cut) to the screws on your new outlet - no need for a new cable or a secondary hole anywhere.

All that said, here's how I would go about accomplishing what you want to do without cutting any unnecessary drywall:

• Turn off the power and remove that outlet in the picture from the box.
• Put a crowbar or something against the inside of the box, and hit it with a hammer a couple of times - just enough to make the box start to lean away from the stud it's attached to so you can see the sides of the nails.
• Slide in the blade of a reciprocating saw and cut off the nails.
• Trace the old cable from the outlet back to the outlet that was feeding it power.
• Disconnect the cable from there and pull it out of the wall if possible. If it's fastened inside the wall, just cut it off and leave it in place.
• Run a new 12-2 cable from your breaker box to the new hole you cut in your drywall for the outlet under your vanity. Run another 12-2 cable from that location to the hole your old outlet box came out of.
• Get a pair of "old work" single gang boxes. These are designed to slide into the small rectangular hole in your drywall and clamp onto it from the back with no need to cut giant holes that will require taping and repair.
• Pull the cables into those boxes and fasten the boxes to the wall.
• Install a new GFCI outlet under the sink using the directions in the package.
• Install a new 15 amp outlet where the old one was located above the sink.

You are exactly right, I use latex caulk to fill the gaps and then paint when dry. This will create a seamless appearance. I've seen homes where the owners have caulked all their trim and it looks fantastic -- very clean.

Edit for clarity: I always strike off the bead square (which I make sure gets pushed down into the space) with a sharp cornered putty knife blade.