Is this an independent or dependent clause?

Solution 1:

First, but goes between conjuncts. The but doesn't go with either clause in this sentence;
rather, it connects the whole sentence with whatever came before it. So I will ignore it here.

Second, the rest of the sentence is an example of the so X/such a(n) X that S construction, which links together an independent clause:

• she was so tired

and a dependent that-clause that identifies her degree of tiredness, referenced to so

• (that) she did not finish painting.

The that complementizer, of course, is optionally deletable.

The entire sentence (without the but) is an answer to

• How tired was she?

Since how is the general wh-word for measuring adjectives and adverbs.