Violin drooping when stopping strings

What you describe is a perfect example of the more extreme side effects of trying to learn without a proper teacher.

Some things I believe could counteract it: sharing the pressure with thumb

That is the normal way it is done. If you had a teacher he/she would have taught you that in the very lesson. In first position the violin should rest half way up the "V" formed by thumb and forefinger.

moving the chin downwards with some extra force

Again, if you had a teacher then you would already know that when you start using phrases like "extra force" that you are describing something dangerous and silly. Applying "force" for any considerable period of time leads to injuries (as you have already discovered). Don't look for more ways to injure yourself.

not pulling the string all the way to the fingerboard

If you had a teacher you would already have learned that the correct motion is not one of pulling the strings down to the fingerboard but of pressing them into the fingerboard from as directly above the strings as you can manage. Yes, the amount of downward pressure will need to vary according to the different musical notes/effects you are trying to produce.

Any thoughts?

Do yourself a favour. Get a real flesh-and-blood teacher then he/she will fix these very basic problems in short order and you will be saved from more pain and suffering. What you are currently doing is ingraining, through much practice, habits which not only lead to injury but which take longer to eradicate the longer you persist.