Hi there my friend,
Congrats on your first LD :bravo: . Anyways, i'll try to answer all of your questions. So here it goes:

Is it normal for a lucid dream to collapse like that, halfway in the dream, halfway in sleep paralysis?
Yeah, a lot of people, including me, experience that feeling of being in the middle of a dream and sleep paralysis. It is often caused by a stabilization issue, Which leads us to your second question.

Also, is there anything I can do to stabilize the dream at that point if I cannot move?
Yeah, there are a lot of ways to stabilize your dream even at the point where you can't move. The trick here is using your mind to stabilize your dream. Shout or command the dream to stabilize through the use of your mind. Do you know this kind of feeling when you day dream? It's as if you're the one moving each part of the character even though you're just day dreaming. it's kinda like that. Just use your mind in stabilizing.

Here's an example: You're lying on the floor, and you cannot move your body. You used your mind and focused it on the nearby door, you tried to make a person come out of the door, help you stand up and assist you in stabilization. A person did came out and everything worked according to plan.

It's a simple yet powerful stabilization technique. Hope this helps