One way or another, I'm convinced that truth does exist. Call it a leap of faith if you will. Faith that there is something that language is describing.
You asked, "If nothing exists, would truth still exist." In the back of my mind, I keep thinking how to phrase things to include the possibility of this being somehow disingenuous (simulated, holographic, imagined, etc.). So I would ask "If what we perceive is not the truth, then would truth still exist." And we're in a similar conundrum to the one presented by the statement "nothing is absolutely true."
Truth at its core transcends just language's attempt to describe it; I believe there is something that language is describing.
Language is all metaphorical, no? The words are (usually) not what the words describe. Utterances fit into the four categories, with a possible fifth being "indeterminable". But the categories themselves, are they arbitrary? I could label them I, II, III, IV, and V...then WHICH category would correspond to truth? Perhaps you could say that the category that includes the statement 2+2=4 is the one that corresponds to truth. Perhaps truth is what language is mapped to in that category.
There is another issue which is proof. Even in math, there are things that are truth that defy proof (a la Godel's Incompleteness theorems). So I seriously doubt that proof is required for something to be true. What constitutes proof is an agreed-upon convention anyway.
So it comes back to what I first said. I am convinced that there is truth, which kind of ties into the power of belief thread and it reminds me of something some famous dead white guy said: Convictions are prisons. But then again they say the truth shall set you free.
Yes, I know...I'm just barely scratching the surface.
"Be as wise as a serpent and as innocent as a dove."