Are the infinitive phrases adverbial or adjectival in these sentences?

Solution 1:

Though they are very similar to adjuncts of purpose, these are probably best analyzed as to-infinitival relative clauses acting as modifiers in a noun phrase headed by water.

These often have modal meaning CaGEL p1068:

Infinitival relatives characteristically have a modal meaning comparable to that expressed in finites by can or should. Here’s something interesting for you to read, for example, is comparable to Here’s something interesting that you can/should read. This modal meaning is indeed what makes relatives ... semantically so close to purpose infinitivals.

Thus,

I am looking for water to drink ___.

is quite similar in meaning to

I am looking for water which I can drink.

and quite different from

I am looking for water to pass the time.

I am looking for water so that I can pass the time.

In your examples there are relativized

1. I am looking for water to drink ___. [object]

2. I am looking for water ___ to quench my thirst. [subject]

3. I am looking for water (for me) to quench my thirst with ____. [object of a preposition]

Comparable relatives for 2 and 3

2. I am looking for water which can quench my thirst.

3. I am looking for water which I can quench my thirst with.

That being said, if one did interpret them as adjuncts, the result would be similar.

1. I am looking for water so that I can drink.

2. I am looking for water so that I can quench my thirst

3. I am looking for water so that I can quench my thirst with (that water).

Though these wouldn't necessarily imply that the water would be the thing that you drink in the first, or the thing that quenches your thirst in the second. Of course we'd have to imagine some scenario where you were required to find water in order to drink something or quench your thirst with something other than the water you found.