I really didn't want to remove the existing elbow from the drain pipe. It is original to my 72 year old house and I didn't want to risk having to get into the wall if something breaks. Coming up with no other solution, I tried to remove it but the nut is absolutely frozen. [i.postimg.cc]

The tail piece from the drain in the new sink is in a different location than the old one. The bottom of the tail piece is about 2-3/4 inches above the bottom of the drain pipe elbow. The center of the elbow is about 5-1/2 inches from the center of the tail piece. That means the trap is about 2 inches short of spanning the distance. It has been about 6 months since I removed the original vanity & sink but I am pretty sure the trap (turned 180 degrees from how I have it in the next photo) was all I needed to connect the tailpiece to the drain elbow. [i.postimg.cc]

The only solution I can think of at this point is to install the trap as shown in the second photo, then use one of those flexible, corrugated plastic pieces to span the distance. Is there another way to go?

Try using penetrating oil on the nut and wait 24 hrs
Chipping away some of the debris all the way around the nut will help a lot too

Pic#2 as you know is not recommended but will work, you may have a slow draining sink because of the deep seal trap

If it were me I would remove the nut and do it properly Even if you have to gently cut into the nut using an oscillating tool and split it away with a flat screwdriver

split the nut carefully with an oscillating tool

replace with a BRASS nut

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"Measure Twice & Cut Once" - Retired U.A. Local 1 & 638

Remove the vanity first so you have plenty of room
And yes, use a brass nut with a rubber washer

yeah, split the old nut and replace with brass.
the new trap arm should easily reach the wall and the most you might have to do is shorten the tailpiece.

As others have said, an oscillating tool with a fine tooth blade will make quick work of that nut.

Before I owned an oscillating tool, I used a propane torch. Those nuts are so large that it is easy to keep the heat on just the nut, and they expand a lot when heated.