Excruciating pain - bad technique?

I'm actually teaching tomorrow, so let me start a bit early.

"Perhaps "intense burning" would have been more appropriate"
Awesome! That's normal,

" ....I voluntarily lied down and took a fall just to curb the pain...."
I call that "every Powder day" now that I'm over 50 ;)

From the description you gave, it's no wonder your quads hurt so much!!! I would not last 15 minutes in that position.

In Canada we teach to our "Technical Reference points"

Use of all joints, be very mobile.
Turning is always led with the lower body.
Edging provides grip and direction change,
We achieve grip through separation of upper and lower body which provides angulation. We call that edging.

If you look at someone with really good separation, it might look as though they are "riding the invisible chair", upon closer examination, you see that the center of mass (balance) is still directly over the outside edge. Or, the head is aligned with the feet, you are not really sitting back. Remember! we do not stand on the downhill ski, we balance on the outside edge, there is a huge difference.

If you achieve good edging through proper separation and balance, you will not skid your turns. The other side effect is that you will feel much more comfortable carrying speed as you will be in control.

I teach this to first timers, intermediates and my friends that consider themselves experts. I just frame it different for each group. The technical reference always stays the same. In "never-ever lessons" these are the basics! the same principals are applied in wedge (used to be called snowplow) turns as in aggressive high speed turns.

"Is it possible to correct this without an instructor" It depends. I don't care what anyone says, you have to be on snow to learn, no amount of reading will make you a better skier. It's all about, mileage !! So, if you have knowledgeable friends that can recognize the deficiencies and help you into a better situation, that's great. Most of us don't have those friends. Sometimes we have friends who think they can do this, try their best with varying degrees of success.

blah blah blah......

Ok, this is important. The goal in skiing is NOT to have great form. It's about having FUN!! That's why we do it. The journey is about having fun. So get good enough that your not killing yourself. Find some like minded friends and have fun! The skills will come!

All this can be achieved on a dry hill on Norfolk (which I have tried when I visited two years ago) and then applied in the Alps! I saw people on the dry hill I could take home and have them look like champs in no time at all.

Guess I got carried away, hope it helped!

For the below, don't get distracted with the level of ability; they are skiing at a pretty aggressive intermediate level, just look at the balance and grip.

Do you stretch your quads and other affected areas before skiing and afterwards?

Your main issue is 100% technique related, do some group lessons. Still absolutely the most fun you can have skiing and guess what watching others both rests your legs and allows you to visualize mistakes until your body can actually feel what is wrong which will take 20+ ski weeks.

As for fitness apparently most of us ski for 45 seconds before stopping. That means when trying to get fit you need to work in this sort of time frame and obviously need to be able to do 4-5 repetitions 1 min apart in order to simulate a ski run.
Few things you could try to test if you are actually there.
Squat with say 50% of body weight on bar for 45 seconds
Spinning flat out at medium torque and 100+ RPM
Skipping rope 45 seconds
etc, etc.