Gödel's Incompleteness Theorem and the undecidability of the halting problem can in fact be proved using each other, so there is good reason to think that there is a deeper connection.
What you suggest isn't actually possible, and to explain why, it's worth stepping back a little bit. Originally, there was a program to formalize mathematics that came to be known as naïve set theory. This theory ran into a major roadblock when Russell proposed a paradox: does the set of all sets that do not contain themselves contain itself? The solution to this, as found in modern set theory, is to very carefully construct sets in such a way that no set can contain itself in the first place, and so the very question isn't expressible in the logic of modern set theory.
The underlying point of both Gödel and Turing is, as you say, constructing a similar statement akin to "this statement is false." But more specifically, what they actually did was to show that, once you meet some relatively low barriers, it is impossible to make a system that omits that construct. In effect, what is done is to encode (respectively) proofs and programs into integers, and then using this encoding, shows that you can always create a self-reference that blows up. Yes, if you change the encoding, it requires a potentially different number to create the self-reference, but the point is that so long as there is a bijection to the natural numbers, then the problematic self-reference must exist.
And, as you may be able to reflect upon, everything we as humans can write or speak can be reduced to a finite-length string of a finite-symbol alphabet, so everything we create must be bijectable with the natural numbers if infinite.